Quantum resource estimation using a re-parameterization method

ABSTRACT

Systems, computer-implemented methods, and computer program products to facilitate estimation of quantum resources to calculate an expectation value of a stochastic process using a re-parameterization method are provided. According to an embodiment, a system can comprise a processor that executes computer executable components stored in memory. The computer executable components can comprise a re-parameterization component that applies a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution to produce a quantum state corresponding to a target probability distribution. The computer executable components can further comprise an estimation component that estimates at least one defined criterion of a quantum computer to be used to compute an expectation value of a stochastic process associated with the target probability distribution.

BACKGROUND

The subject disclosure relates to estimation of quantum resources tocalculate an expectation value of a stochastic process, and morespecifically, to estimation of quantum resources to calculate anexpectation value of a stochastic process using a re-parameterizationmethod.

SUMMARY

The following presents a summary to provide a basic understanding of oneor more embodiments of the invention. This summary is not intended toidentify key or critical elements, or delineate any scope of theparticular embodiments or any scope of the claims. Its sole purpose isto present concepts in a simplified form as a prelude to the moredetailed description that is presented later. In one or more embodimentsdescribed herein, systems, devices, computer-implemented methods, and/orcomputer program products that can facilitate estimation of quantumresources to calculate an expectation value of a stochastic processusing a re-parameterization method are described.

According to an embodiment, a system can comprise a processor thatexecutes computer executable components stored in memory. The computerexecutable components can comprise a re-parameterization component thatapplies a quantum fault-tolerant operation to a variationally preparedquantum state corresponding to a probability distribution to produce aquantum state corresponding to a target probability distribution. Thecomputer executable components can further comprise an estimationcomponent that estimates at least one defined criterion of a quantumcomputer to be used to compute an expectation value of a stochasticprocess associated with the target probability distribution.

According to another embodiment, a computer-implemented method cancomprise applying, by a system operatively coupled to a processor, aquantum fault-tolerant operation to a variationally prepared quantumstate corresponding to a probability distribution to produce a quantumstate corresponding to a target probability distribution. Thecomputer-implemented method can further comprise estimating, by thesystem, at least one defined criterion of a quantum computer to be usedto compute an expectation value of a stochastic process associated withthe target probability distribution.

According to another embodiment, a computer program product comprising acomputer readable storage medium having program instructions embodiedtherewith, the program instructions executable by a processor to causethe processor to apply a quantum fault-tolerant operation to avariationally prepared quantum state corresponding to a probabilitydistribution to produce a quantum state corresponding to a targetprobability distribution. The program instructions are furtherexecutable by the processor to cause the processor to estimate at leastone defined criterion of a quantum computer to be used to compute anexpectation value of a stochastic process associated with the targetprobability distribution.

DESCRIPTION OF THE DRAWINGS

FIGS. 1, 2, and 3 illustrate block diagrams of example, non-limitingsystems that can each facilitate estimation of quantum resources tocalculate an expectation value of a stochastic process using are-parameterization method in accordance with one or more embodimentsdescribed herein.

FIGS. 4, 6A, 6B, 6C, 7A, 7B, 8A, 8B, 9A, 9B, and 9C illustrate example,non-limiting graphs that can facilitate estimation of quantum resourcesto calculate an expectation value of a stochastic process using are-parameterization method in accordance with one or more embodimentsdescribed herein.

FIG. 5 illustrates an example, non-limiting circuit that can facilitateestimation of quantum resources to calculate an expectation value of astochastic process using a re-parameterization method in accordance withone or more embodiments described herein.

FIG. 10 illustrates a flow diagram of an example, non-limitingcomputer-implemented methods that can facilitate estimation of quantumresources to calculate an expectation value of a stochastic processusing a re-parameterization method in accordance with one or moreembodiments described herein.

FIG. 11 illustrates a block diagram of an example, non-limitingoperating environment in which one or more embodiments described hereincan be facilitated.

FIG. 12 illustrates a block diagram of an example, non-limiting cloudcomputing environment in accordance with one or more embodiments of thesubject disclosure.

FIG. 13 illustrates a block diagram of example, non-limiting abstractionmodel layers in accordance with one or more embodiments of the subjectdisclosure.

DETAILED DESCRIPTION

The following detailed description is merely illustrative and is notintended to limit embodiments and/or application or uses of embodiments.Furthermore, there is no intention to be bound by any expressed orimplied information presented in the preceding Background or Summarysections, or in the Detailed Description section.

One or more embodiments are now described with reference to thedrawings, wherein like referenced numerals are used to refer to likeelements throughout. In the following description, for purposes ofexplanation, numerous specific details are set forth in order to providea more thorough understanding of the one or more embodiments. It isevident, however, in various cases, that the one or more embodiments canbe practiced without these specific details.

As referenced herein, a “derivative” and/or a “derivative asset” is acontract between an issuer and a holder, which is valid until itsexpiration date. As referenced herein, an “entity” can comprise a human,a client, a user, a computing device, a software application, an agent,a machine learning (ML) model, an artificial intelligence (AI), and/oranother entity. It will be understood that when an element is referredto herein as being “coupled” to another element, it can describe one ormore different types of coupling including, but not limited to, chemicalcoupling, communicative coupling, electrical coupling, electromagneticcoupling, operative coupling, optical coupling, physical coupling,thermal coupling, and/or another type of coupling.

FIGS. 1, 2, and 3 illustrate block diagrams of example, non-limitingsystems 100, 200, and 300, respectively, that can each facilitateestimation of quantum resources to calculate an expectation value of astochastic process using a re-parameterization method in accordance withone or more embodiments described herein. System 100, 200, and 300 caneach comprise a quantum resource estimation system 102. Quantum resourceestimation system 102 of system 100 depicted in FIG. 1 can comprise amemory 104, a processor 106, a re-parameterization component 108, anestimation component 110, and/or a bus 112. Quantum resource estimationsystem 102 of system 200 depicted in FIG. 2 can further comprise avariational component 202. Quantum resource estimation system 102 ofsystem 300 depicted in FIG. 3 can further comprise an error analysiscomponent 302.

Although some embodiments of the subject disclosure describe an exampleapplication of quantum resource estimation system 102 to estimatequantum resources to calculate an expectation value of a stochasticprocess such as, for instance, a derivative asset, it should beappreciated that the subject disclosure is not so limiting. For example,quantum resource estimation system 102 can estimate quantum resources tocalculate an expectation value of another stochastic process (e.g., anytype of stochastic process).

It should be appreciated that the embodiments of the subject disclosuredepicted in various figures disclosed herein are for illustration only,and as such, the architecture of such embodiments are not limited to thesystems, devices, and/or components depicted therein. For example, insome embodiments, system 100, system 200, system 300, and/or quantumresource estimation system 102 can further comprise various computerand/or computing-based elements described herein with reference tooperating environment 1100 and FIG. 11. In several embodiments, suchcomputer and/or computing-based elements can be used in connection withimplementing one or more of the systems, devices, components, and/orcomputer-implemented operations shown and described in connection withFIG. 1, FIG. 2, FIG. 3, and/or other figures disclosed herein.

Memory 104 can store one or more computer and/or machine readable,writable, and/or executable components and/or instructions that, whenexecuted by processor 106 (e.g., a classical processor, a quantumprocessor, and/or another type of processor), can facilitate performanceof operations defined by the executable component(s) and/orinstruction(s). For example, memory 104 can store computer and/ormachine readable, writable, and/or executable components and/orinstructions that, when executed by processor 106, can facilitateexecution of the various functions described herein relating to quantumresource estimation system 102, re-parameterization component 108,estimation component 110, variational component 202, error analysiscomponent 302, and/or another component associated with quantum resourceestimation system 102 as described herein with or without reference tothe various figures of the subject disclosure.

Memory 104 can comprise volatile memory (e.g., random access memory(RAM), static RAM (SRAM), dynamic RAM (DRAM), and/or another type ofvolatile memory) and/or non-volatile memory (e.g., read only memory(ROM), programmable ROM (PROM), electrically programmable ROM (EPROM),electrically erasable programmable ROM (EEPROM), and/or another type ofnon-volatile memory) that can employ one or more memory architectures.Further examples of memory 104 are described below with reference tosystem memory 1116 and FIG. 11. Such examples of memory 104 can beemployed to implement any embodiments of the subject disclosure.

Processor 106 can comprise one or more types of processors and/orelectronic circuitry (e.g., a classical processor, a quantum processor,and/or another type of processor and/or electronic circuitry) that canimplement one or more computer and/or machine readable, writable, and/orexecutable components and/or instructions that can be stored on memory104. For example, processor 106 can perform various operations that canbe specified by such computer and/or machine readable, writable, and/orexecutable components and/or instructions including, but not limited to,logic, control, input/output (I/O), arithmetic, and/or the like. In someembodiments, processor 106 can comprise one or more central processingunit, multi-core processor, microprocessor, dual microprocessors,microcontroller, System on a Chip (SOC), array processor, vectorprocessor, quantum processor, and/or another type of processor. Furtherexamples of processor 106 are described below with reference toprocessing unit 1114 and FIG. 11. Such examples of processor 106 can beemployed to implement any embodiments of the subject disclosure.

Quantum resource estimation system 102, memory 104, processor 106,re-parameterization component 108, estimation component 110, variationalcomponent 202, error analysis component 302, and/or another component ofquantum resource estimation system 102 as described herein can becommunicatively, electrically, operatively, and/or optically coupled toone another via bus 112 to perform functions of system 100, system 200,system 300, quantum resource estimation system 102, and/or anycomponents coupled therewith. Bus 112 can comprise one or more memorybus, memory controller, peripheral bus, external bus, local bus, aquantum bus, and/or another type of bus that can employ various busarchitectures. Further examples of bus 112 are described below withreference to system bus 1118 and FIG. 11. Such examples of bus 112 canbe employed to implement any embodiments of the subject disclosure.

Quantum resource estimation system 102 can comprise any type ofcomponent, machine, device, facility, apparatus, and/or instrument thatcomprises a processor and/or can be capable of effective and/oroperative communication with a wired and/or wireless network. All suchembodiments are envisioned. For example, quantum resource estimationsystem 102 can comprise a server device, a computing device, ageneral-purpose computer, a special-purpose computer, a quantumcomputing device (e.g., a quantum computer), a tablet computing device,a handheld device, a server class computing machine and/or database, alaptop computer, a notebook computer, a desktop computer, a cell phone,a smart phone, a consumer appliance and/or instrumentation, anindustrial and/or commercial device, a digital assistant, a multimediaInternet enabled phone, a multimedia players, and/or another type ofdevice.

Quantum resource estimation system 102 can be coupled (e.g.,communicatively, electrically, operatively, optically, and/or coupledvia another type of coupling) to one or more external systems, sources,and/or devices (e.g., classical and/or quantum computing devices,communication devices, and/or another type of external system, source,and/or device) using a wire and/or a cable. For example, quantumresource estimation system 102 can be coupled (e.g., communicatively,electrically, operatively, optically, and/or coupled via another type ofcoupling) to one or more external systems, sources, and/or devices(e.g., classical and/or quantum computing devices, communicationdevices, and/or another type of external system, source, and/or device)using a data cable including, but not limited to, a High-DefinitionMultimedia Interface (HDMI) cable, a recommended standard (RS) 232cable, an Ethernet cable, and/or another data cable.

In some embodiments, quantum resource estimation system 102 can becoupled (e.g., communicatively, electrically, operatively, optically,and/or coupled via another type of coupling) to one or more externalsystems, sources, and/or devices (e.g., classical and/or quantumcomputing devices, communication devices, and/or another type ofexternal system, source, and/or device) via a network. For example, sucha network can comprise wired and/or wireless networks, including, butnot limited to, a cellular network, a wide area network (WAN) (e.g., theInternet), a local area network (LAN), and/or another network. Quantumresource estimation system 102 can communicate with one or more externalsystems, sources, and/or devices, for instance, computing devices usingvirtually any desired wired and/or wireless technology, including butnot limited to: wireless fidelity (Wi-Fi), global system for mobilecommunications (GSM), universal mobile telecommunications system (UMTS),worldwide interoperability for microwave access (WiMAX), enhancedgeneral packet radio service (enhanced GPRS), third generationpartnership project (3GPP) long term evolution (LTE), third generationpartnership project 2 (3GPP2) ultra mobile broadband (UMB), high speedpacket access (HSPA), Zigbee and other 802.XX wireless technologiesand/or legacy telecommunication technologies, BLUETOOTH®, SessionInitiation Protocol (SIP), ZIGBEE®, RF4CE protocol, WirelessHARTprotocol, 6LoWPAN (IPv6 over Low power Wireless Area Networks), Z-Wave,an ANT, an ultra-wideband (UWB) standard protocol, and/or otherproprietary and non-proprietary communication protocols. Therefore, insome embodiments, quantum resource estimation system 102 can comprisehardware (e.g., a central processing unit (CPU), a transceiver, adecoder, quantum hardware, a quantum processor, and/or other hardware),software (e.g., a set of threads, a set of processes, software inexecution, quantum pulse schedule, quantum circuit, quantum gates,and/or other software) or a combination of hardware and software thatcan facilitate communicating information between quantum resourceestimation system 102 and external systems, sources, and/or devices(e.g., computing devices, communication devices, and/or another type ofexternal system, source, and/or device).

Quantum resource estimation system 102 can comprise one or more computerand/or machine readable, writable, and/or executable components and/orinstructions that, when executed by processor 106 (e.g., a classicalprocessor, a quantum processor, and/or another type of processor), canfacilitate performance of operations defined by such component(s) and/orinstruction(s). Further, in numerous embodiments, any componentassociated with quantum resource estimation system 102, as describedherein with or without reference to the various figures of the subjectdisclosure, can comprise one or more computer and/or machine readable,writable, and/or executable components and/or instructions that, whenexecuted by processor 106, can facilitate performance of operationsdefined by such component(s) and/or instruction(s). For example,re-parameterization component 108, estimation component 110, variationalcomponent 202, error analysis component 302, and/or any other componentassociated with quantum resource estimation system 102 as disclosedherein (e.g., communicatively, electronically, operatively, and/oroptically coupled with and/or employed by quantum resource estimationsystem 102), can comprise such computer and/or machine readable,writable, and/or executable component(s) and/or instruction(s).Consequently, according to numerous embodiments, quantum resourceestimation system 102 and/or any components associated therewith asdisclosed herein, can employ processor 106 to execute such computerand/or machine readable, writable, and/or executable component(s) and/orinstruction(s) to facilitate performance of one or more operationsdescribed herein with reference to quantum resource estimation system102 and/or any such components associated therewith.

Quantum resource estimation system 102 can facilitate (e.g., viaprocessor 106) performance of operations executed by and/or associatedwith re-parameterization component 108, estimation component 110,variational component 202, error analysis component 302, and/or anothercomponent associated with quantum resource estimation system 102 asdisclosed herein. For example, as described in detail below, quantumresource estimation system 102 can facilitate (e.g., via processor 106):applying a quantum fault-tolerant operation to a variationally preparedquantum state corresponding to a probability distribution to produce aquantum state corresponding to a target probability distribution; and/orestimating at least one defined criterion of a quantum computer to beused to compute an expectation value of a stochastic process associatedwith the target probability distribution.

In another example, as described in detail below, quantum resourceestimation system 102 can further facilitate (e.g., via processor 106):applying a transformation operation to the variationally preparedquantum state to produce the quantum state corresponding to the targetprobability distribution with a defined mean of the target probabilitydistribution, a defined standard deviation of the target probabilitydistribution, and/or one or more explicit parameters that specify thetarget probability distribution; applying the quantum fault-tolerantoperation to the variationally prepared quantum state to prepare thequantum state as a superposition over possible paths of a discrete timemultivariate stochastic process; training a variational quantum circuitto prepare the variationally prepared quantum state and to reducecomputational costs of quantum arithmetic operations performed by thequantum computer to compute the expectation value of the stochasticprocess associated with the target probability distribution; trainingthe variational quantum circuit using a Hamiltonian operator to generatea ground state corresponding to the target probability distribution;calculating one or more errors associated with at least one of:application of the quantum fault-tolerant operation to the variationallyprepared quantum state to produce the quantum state, estimation of theat least one defined criterion, or computation of the expectation valueof the stochastic process associated with the target probabilitydistribution.

In the above examples, the at least one defined criterion can comprisean attribute, a condition, a property, a parameter, and/or aconfiguration of the quantum computer that enables the quantum computerto achieve a defined quantum advantage in computing the expectationvalue of the stochastic process associated with the target probabilitydistribution. In the above examples, the probability distribution cancomprise a standard normal probability distribution and the targetprobability distribution can comprise a normal probability distribution.

In accordance with various embodiments of the subject disclosuredescribed herein, to perform one or more of the above describedoperations, quantum resource estimation system 102, re-parameterizationcomponent 108, estimation component 110, variational component 202,and/or error analysis component 302 can define and/or implement one ormore of the algorithms (e.g., Algorithm 2.1, 3.1, 3.2, 4.1, and/or 4.2)and/or one or more of the equations (e.g., Equations (1)-(109))described below with reference to Sections 1.0-11.0. For example, withreference to Sections 3.2 and 10.0 described below, re-parameterizationcomponent 108 can apply a quantum fault-tolerant operation to avariationally prepared quantum state corresponding to a probabilitydistribution (e.g., a standard normal probability distribution) toproduce a quantum state corresponding to a target probabilitydistribution (e.g., a normal probability distribution). In this example,re-parameterization component 108 can apply the quantum fault-tolerantoperation to the variationally prepared quantum state to prepare thequantum state as a superposition over possible paths of a discrete timemultivariate stochastic process. In this example, re-parameterizationcomponent 108 can apply a transformation operation to the variationallyprepared quantum state to produce the quantum state corresponding to thetarget probability distribution with at least one of: a defined mean ofthe target probability distribution; a defined standard deviation of thetarget probability distribution; or one or more explicit parameters thatspecify the target probability distribution.

In another example, with reference to Sections 3.1.2 and 3.2.3 describedbelow, estimation component 110 can estimate at least one definedcriterion of a quantum computer to be used to compute an expectationvalue of a stochastic process associated with the target probabilitydistribution. For instance, estimation component 110 can estimate atleast one defined criterion of a quantum computer to be used to computea value of an asset such as, for example, a derivative asset associatedwith the target probability distribution. For example, estimationcomponent 110 can estimate at least one defined criterion including, butnot limited to, an attribute, a condition, a property, a parameter, aconfiguration, and/or another criterion of the quantum computer thatenables the quantum computer to achieve a defined quantum advantage incomputing an expectation value of a stochastic process (e.g., the valueof a derivative asset) associated with the target probabilitydistribution.

In another example, with reference to Sections 3.2.1 and 11.0 describedbelow, variational component 202 can train a variational quantum circuitto prepare the variationally prepared quantum state. For instance,variational component 202 can train a variational quantum circuit toprepare the variationally prepared quantum state by training thevariational quantum circuit using a Hamiltonian operator to generate aground state corresponding to the target probability distribution. Itshould be appreciated that variational component 202 can train avariational quantum circuit to prepare the variationally preparedquantum state to reduce computational costs of quantum arithmeticoperations performed by the quantum computer to compute the expectationvalue of the stochastic process (e.g., the value of a derivative asset)associated with the target probability distribution.

In another example, with reference to Section 3.2.2 described below,error analysis component 302 can calculate one or more errors associatedwith at least one of: application of the quantum fault-tolerantoperation to the variationally prepared quantum state to produce thequantum state; estimation of the at least one defined criterion; orcomputation of the expectation value of the stochastic process (e.g.,the value of a derivative asset) associated with the target probabilitydistribution.

In accordance with various embodiments and as described in the followingsections, quantum resource estimation system 102 can determine an upperbound on the resources (e.g., quantum computing resources) involved toprovide a valuable quantum advantage in pricing derivatives. To do so,quantum resource estimation system 102 can provide the first completeresource estimates for useful quantum derivative pricing, usingautomatically callable (auto-callable) option and a target accrualredemption forwards (TARF) derivatives as example benchmark use cases.Quantum resource estimation system 102 can overcome blocking challengesin the known approaches and provide a new method for quantum derivativepricing—the re-parameterization method—that avoids them. It should beappreciated that the re-parameterization method that can be definedand/or implemented by quantum resource estimation system 102 asdescribed in the sections below hybridizes pre-trained variationalcircuits with fault-tolerant quantum computing to dramatically reduceresources (e.g., quantum computing resources) involved with estimatingthe value of a derivative. As described below and in accordance withvarious embodiments of the subject disclosure, quantum resourceestimation system 102 can determine that the auto-callable option andTARF derivatives as example benchmark use cases involve, for instance:approximately 8,000 (8 k) logical qubits; an approximate T-depth of 50million; and an estimated logical clock speed of approximately 10megahertz (MHz) to achieve a defined quantum advantage.

1.0 Derivative Pricing

The pricing of derivative contracts using Monte Carlo methods consumessignificant computation in the financial sector, and quantum advantagein this application would be very valuable. In accordance with variousembodiments and as described in the following sections, quantum resourceestimation system 102 can provide the first detailed resource estimatesof the conditions involved with achieving a quantum advantage inderivative pricing. To accomplish this, quantum resource estimationsystem 102 can define and/or implement the new methods described belowin accordance with one or more embodiments of the subject disclosure toload stochastic processes into a quantum computer.

As defined above and as referenced herein, a “derivative” and/or a“derivative asset” is a contract between an issuer and a holder, whichis valid until its expiration date. Examples of these derviatives assetsinclude, but are not limited to, a forward contract, an option, anauto-callable option, a target accrual redemption note (TARN), a TARF,and/or another derivative asset. Each derivative defines a payoff thatdefines what the holder stands to gain. Payoffs depend on the value ofone or more underlying assets across the duration of the contract.Examples of these underlying assets include, but are not limited to,stocks, currencies, commodities, and/or another underlying asset.Derivative contracts are ubiquitous in the finance domain with varioususes from hedging risk to speculation. The goal of derivative pricing isto determine the value of entering the derivative contract today, givenuncertainty about future values of the underlying assets andconsequently the payoff.

The underlying asset is typically modeled as a stochastic process underassumptions like no-arbitrage. No-arbitrage is the assumption that nospecific asset is priced differently in different marketplaces such thatone can never buy an asset from one marketplace and immediately sell itat another for a profit. A common model is that the underlying assetsevolve under geometric Brownian motion. Let {right arrow over (S)}^(t)ϵ

₊ ^(d) be a vector of values for d underlyings at time t. Let ({rightarrow over (S)}⁰, . . . , {right arrow over (S)}^(T))=ωϵΩ be a path of adiscrete time multivariate stochastic process describing the values ofthose assets. Both notations are used herein for a path. Thecorresponding probability density function is denoted by p(ω). Letf(ω)=f({right arrow over (S)}⁰, . . . , {right arrow over (S)}^(T))ϵ

be the payoff of some derivative on those assets. Equation (1) definedbelow can be used to price the derivative.

(f)=∫ _(ωϵΩ) p (ω)f(ω)dω.  (1)

If the underlying stochastic processes are modeled with geometricBrownian motion then they have transition probabilities

$\begin{matrix}{{P\left( {{\overset{\rightarrow}{S}}^{t}❘{\overset{\rightarrow}{S}}^{t - 1}} \right)} = {\frac{\exp\left( {{- \frac{1}{2}}\left( {{\ln{\overset{\rightarrow}{S}}^{t}} - {\overset{\rightarrow}{\mu}}^{t - 1}} \right)^{T}{\sum^{- 1}\left( {{\ln{\overset{\rightarrow}{S}}^{t}} - {\overset{\rightarrow}{\mu}}^{t - 1}} \right)}} \right)}{\left( {2\pi} \right)^{d/2}\left( {\det\sum} \right)^{1/2}{\prod_{j = 1}^{d}S_{j}^{t}}}.}} & (2) \\{where} & \; \\{{{\ln{\overset{\rightarrow}{S}}^{t}} = \left( {{\ln S_{1}^{t}},{\ln S_{2}^{t}},\ldots\mspace{14mu},{\ln\; S_{a}^{t}}} \right)^{T}}{{\overset{\rightarrow}{\mu}}^{t - 1} = \left( {\mu_{1}^{t - 1},\mu_{2}^{t - 1},\ldots\mspace{14mu},\mu_{d}^{t - 1}} \right)^{T}}{\mu_{j}^{t - 1} = {{\left( {r - {{0.5}\sigma_{j}^{2}}} \right)\Delta t} + {\ln{S_{j}^{t - 1}.}}}}} & (3)\end{matrix}$

Note that Equation (2) defined above at time t has a dependency on theasset vector at time t−1 via ln S_(j) ^(−t−1) in μ_(j) ^(t−1). Theparameters r and σ_(j) are the risk-free rate and the volatility of thej-th asset respectively, Δt is the time duration between steps of thestochastic process, and Σ is the d×d positive-definite covariance matrixof the d underlyings

$\begin{matrix}{\sum{= {\Delta\;{t\begin{bmatrix}\sigma_{1}^{2} & {\rho_{12}\sigma_{1}\sigma_{2}} & \ldots & {\rho_{1d}\sigma_{1}\sigma_{d}} \\{\rho_{21}\sigma_{2}\sigma_{1}} & \sigma_{2}^{2} & \ldots & {\rho_{2d}\sigma_{2}\sigma_{d}} \\\vdots & \vdots & \ddots & \vdots \\{\rho_{d\; 1}\sigma_{d}\sigma_{1}} & \ldots & \ldots & \sigma_{d}^{2}\end{bmatrix}}}}} & (4)\end{matrix}$

where −1≤ρ_(ij)≤1 is the correlation between assets i and j. Theprobability of any particular path ωϵΩ is then

p (ω)=Π_(t=1) ^(T) P({right arrow over (S)} ^(t) |{right arrow over (S)}_(t-1)).  (5)

The risk-free rate referenced above is the rate of return of investingin a risk-free asset. Although such an asset is purely theoretical,treasury bonds are typically used to represent such an asset andapproximate r as the yield of the treasury bond minus the currentinflation rate.

Classically some simple derivatives under this model are easy to price,such as European call options that can be priced analytically using theBlack-Scholes equation. Easy to price derivatives are often pathindependent, where the payoff is only a function of the final prices atexercise time f(ω)=f(S ^(T)). This contrasts with path dependentderivatives that are more difficult to price. Path dependent derivativesare often priced in practice with classical Monte Carlo methods.

Using classical Monte Carlo, the accuracy of derivative pricingconverges as O(1/√{square root over (M)}), where M is the number ofsamples. In general cases, quantum algorithms based on amplitudeestimation can be used to improve this to O(1/M). Recent work hasconsidered how to specialize this advantage to options pricing and riskanalysis. As this is only a quadratic speedup, the subject disclosurefocuses on derivatives that are complicated enough to have a large M inpractice. In accordance with various embodiments and as described in thefollowing sections, quantum resource estimation system 102 can provideend to end quantum resource estimates for two examples of suchderivatives, auto-callable options and TARFs, which are bothcomputationally expensive, path-dependent derivatives. In doing so,quantum resource estimation system 102 (e.g., via re-parameterizationcomponent 108, estimation component 110, and/or variational component202) can detail and optimize the loading into quantum states of theunderlying distribution over asset paths. This loading step was leftopen (e.g., unresolved) in prior art technologies and quantum resourceestimation system 102 (e.g., via estimation component 110) can providethe first account of the resources involved to accomplish such a loadingstep.

In addition to estimating the resources that can be used for pathloading, quantum resource estimation system 102 can also provide severaloptimizations, including intentional shifts from price space to returnspace calculations and the new re-parameterization method. These methodsreduce the resources significantly and are summarized in Table 1.

(d, T) Error T-count T-depth # Logical Qubits Method Auto TARF Auto TARFAuto TARF Auto TARF Auto TARF Riemann Sum (3, 20) (1, 26) 2 × 10⁻³ ≥10¹³≥10¹⁸ ≥10¹³ ≥10¹⁸ — — Riemann Sum (no-norm) 1.4 × 10¹¹ 5.5 × 10¹⁰ 1.9 ×10⁸ 1.7 × 10⁸  24 k  15 k Re-parameterization 4.2 × 10⁹   3 × 10⁹ 4.6 ×10⁷ 6.2 × 10⁷ 7.5 k 9.5 k

Table 1 depicted above illustrates the resources estimated byimplementing quantum resource estimation system 102 in accordance withone or more embodiments of the subject disclosure to price derivativesusing different methods for a target error of 2×10⁻³. In suchimplementation of quantum resource estimation system 102, a basket ofauto-callable (auto) options with 5 auto-call dates and a knock-in putoption, and a TARF with one underlying and 26 simulation dates. In suchimplementation of quantum resource estimation system 102, it wasdetermined that Grover-Rudolph methods are not applicable in practiceand that Riemann summation methods can involve normalization assumptionsto avoid errors that grow exponentially in T. Even if thosenormalization issues were avoided, as detailed in the Riemann Sum(no-norm) row of Table 1, the re-parameterization method that can bedefined and/or implemented by quantum resource estimation system 102 inaccordance with one or more embodiments of the subject disclosureperforms best. Section 3.1 below describes the Riemann summationnormalization and the detailed resource estimation that can be definedby and/or implemented by estimation component 110 is discussed in belowin Sections 3.1.2 and 3.2.3.

1.1 Discretized Derivative Pricing

In order to map a derivative pricing problem into quantum states, thevalues {right arrow over (S)}^(t) are discretized. Classically, this isnot that important as high precision is available, but in order to studythe minimal qubit criteria, discretization is explicitly considered.

Let each value {right arrow over (S)}^(t) be discretized into adifferent n-qubit registers, that is, mapped to a regular grid. Thediscrete space of paths can then be defined as ωϵΩ. The priceexpectation is now a sum

(f)=Σ_(ψϵΩ) p(ω)f(ω),  (6)

where the probability p(ω) can be defined in multiple ways. Forinstance, the midpoints of the grid cells can be taken so that

p(ω)=Π_(t=1) ^(T) P({right arrow over (S)} ^(t) |S ^(t−1)),  (7)

where the S ^(t) are restricted to discrete midpoints. Or p(ω) can bedefined as an integral over the discrete cells. These representationsare the same in the limit of fine grids and in accordance with one ormore embodiments of the subject disclosure, the midpoint method is used.

1.2 Price Space vs. Return Space

As described above, geometric Brownian motion can be used to model theprice on underlying assets. As referenced herein, this is called theprice space description of the underlying stochastic process. In pricespace, transition probabilities are given by a multivariate log-normaldistribution.

An alternative, but equivalent representation, is to consider thestochastic process on the log-returns of the underlying assets, andperform all calculations in return space. When asset prices obey alog-normal distribution, then the log-returns are distributed normally.A vector of underlying log-returns for d assets at time t can be definedas R ^(t)=(R₁ ^(t), R₂ ^(t), . . . , R_(d) ^(t)). The transitionprobabilities can then be given by a multivariate normal distribution

$\begin{matrix}{{{P\left( {\overset{\rightarrow}{R}}^{t} \right)} = \frac{\exp\left( {{- \frac{1}{2}}\left( {R^{t} - \overset{\rightarrow}{\mu}} \right)^{T}{\sum^{- 1}\left( {R^{t} - \overset{\rightarrow}{\mu}} \right)}} \right)}{\left( {2\pi} \right)^{d/2}\left( {\det\sum} \right)^{1/2}}},} & (8) \\{{where},} & \; \\{{\mu = \left( {\mu_{1},\mu_{2},{\ldots\mspace{14mu}\mu_{a}}} \right)^{T}},} & (9) \\{{\mu_{j} = {\left( {r - {{0.5}\sigma_{j}^{2}}} \right)\Delta\; t}},} & (10)\end{matrix}$

and σ, Δt, Σ, and r are the same Brownian motion parameters as in pricespace. Note that this is no longer conditioned on the value at theprevious time step. In fact, the path distribution in return spaceincludes dT independent Gaussians.

Note the overloaded notation from the price space formulation as theserepresentations are interchangeable. At any time t′ the price of asset jcan be calculated from return space using

$\begin{matrix}{S_{j}^{t^{\prime}} = {S_{j}^{0}{\prod_{t = 1}^{t^{\prime}}{e^{R_{j}^{t}}.}}}} & (11)\end{matrix}$

This calculation is used when the stochastic process has been modelledin return space but the payoff is defined in terms of asset prices. Inthe various embodiments of the subject disclosure described herein, itwill be made clear from the context which space such embodiments areoperating in.

Switching between price space and return space changes from log-normaldistribution loading to normal distribution loading. In general, theloading of normals is easier since their underlying stochastic evolutionis independent of the price at a previous time step which can be seen bycomparing Equation (3) and Equation (10) defined above. As such, theprobability distribution P({right arrow over (R)}¹, {right arrow over(R)}², . . . , {right arrow over (R)}^(T)) across all T timesteps of thestochastic process can be computed simultaneously with

$\begin{matrix}{{{P\left( \overset{\rightarrow}{R} \right)} \equiv {P\left( {{\overset{\rightarrow}{R}}^{1},{\overset{\rightarrow}{R}}^{2},\ldots\mspace{14mu},{\overset{\rightarrow}{R}}^{T}} \right)}} = {\frac{\exp\left( {\sum_{t = 1}^{T}{{- \frac{1}{2}}\left( {{\overset{\rightarrow}{R}}^{t} - \overset{\rightarrow}{\mu}} \right){\sum^{- 1}\left( {{\overset{\rightarrow}{R}}^{t} - \overset{\rightarrow}{\mu}} \right)}}} \right)}{\left( {2\pi} \right)^{d{T/2}}\left( {\det\sum} \right)^{T/2}}.}} & (12)\end{matrix}$

This advantage can compensate for the quantum arithmetic used toevaluate the exponentials in Equation (11) defined above. In variousembodiments of the subject disclosure, quantum resource estimationsystem 102 can leverage this advantage by using the re-parameterizationmethod described herein. Additionally, when working with derivativesthat have payoffs defined in terms of log-returns directly and areindependent of individual asset prices, this is another reason quantumresource estimation system 102 can work in return space.

2.0 Core Approach

The approach of the subject disclosure to derivative pricing extends thequantum mean estimation method. Let the normalized payoff of any path begiven by

$\begin{matrix}{{\overset{\sim}{f}(\omega)} = {\frac{{f(\omega)} - f_{\min}}{f_{\max} - f_{\min}} \in {\left\lbrack {0,1} \right\rbrack.}}} & (13)\end{matrix}$

Algorithm 2.1 defined below can proceed in four phases. First, aprobability distribution is loaded in the form of a superposition overall possible paths. Second, payoffs for all possible paths arecalculated in quantum parallel. Third, the expected payoff is stored inthe amplitude of a marked state. Fourth, amplitude estimation is used toread out the amplitude using

(1/ϵ) queries for a given target accuracy ϵ>0.

Algorithm 2.1—Core Approach to Derivative Pricing

Use parameters n, d, and T that are all positive integers.

Obtain an operator

for loading a probabilistically weighted superposition of paths onto aregister of ndT-qubits.

1. Apply operator

to prepare the quantum state

|0

=Σ_(ω)√{square root over (p(ω))}|ω

.  (14)

2. Calculate δ(ω)=arcsin √{square root over ({acute over (f)}(ω))} intoa quantum register

Σ_(ω)√{square root over (p(ω))}|ω

|δ(ω)

.  (15)

3. Introduce an ancilla qubit and rotate the value of the {tilde over(f)}(ω) register into its amplitude:

Σ_(ω)√{square root over (p(ω)(1−{tilde over (f)}(ω)))}|ω

|0

+Σ_(ω)√{square root over (p(ω){tilde over (f)}(ω))}|ω

|1

.  (16)

4. Use amplitude estimation to extract the probability of the Ancillabeing 1), which is the

(e.g., discretized) expected payoff

({tilde over (f)}). Rescale this to obtain

(f)=(f_(max)−f_(min))

({tilde over (f)})+f_(min).

Note that Steps 1-3 in the Algorithm 2.1 load the exact answer after asingle execution. Were it possible to read out an amplitude directly,then quantum resource estimation system 102 could compute theexpectation over all paths in a constant number of queries. This is,unfortunately, not possible, and so amplitude estimation introduces alinear overhead to extract an answer to a given precision. This can be akey conceptual difference from classical Monte Carlo where samples frompaths are taken. In Algorithm 2.1, quantum resource estimation system102 can compute (e.g. all) possible paths and take (e.g., amplitudeestimated) samples of the expected payoff.

Another distinguishing feature of the quantum approach is that quantumresource estimation system 102 can normalize the payoff in order tostore it in the amplitude of a state. This normalization can be rescaledat the end and can have an impact on error scaling, as errors are alsoscaled up. In the Riemann summation method, discussed in Section 3.1, aversion of this normalization rescaling can rapidly accumulate errors.

2.1 Amplitude Estimation for Derivative Pricing

Typically, path-dependent derivatives like auto-callables and TARFs arepriced using Monte Carlo methods. Paths ω=({right arrow over (S)}⁰,{right arrow over (S)}¹, . . . , {right arrow over (S)}^(T)) aregenerated by modeling the underlying stochastic process and then theexpected payoff is calculated using the estimator

$\begin{matrix}{{{\mathbb{E}}(f)} \approx {\frac{1}{M}{\sum_{\omega = 1}^{M}{{f(\omega)}.}}}} & (17)\end{matrix}$

This estimator converges to the true expected value with errorϵ=0(M^(−1/2)) by the Central Limit Theorem.

This convergence can be quadratically accelerated to ϵ=O(M⁻¹) usingquantum amplitude estimation for Monte Carlo. Amplitude estimation takesas input a unitary operator

on n+1 qubits such that

|0

_(n+1)=√{square root over (1−a)}|ψ₀

_(n)|0

+√{square root over (a)}|ψ₁

|1

,  (18)

where the parameter a is unknown. Here the final qubit acts as a labelto distinguish |ω₀

states from |ψ₁

states.

Amplitude estimation determines a by repeated applications of theoperator (often referred to as the Grover operator)

=

S₀

^(†)S_(ψ) ₀ , where S₀=

−2|0

_(n+1)

0|_(n+1) and S_(ψ) ₀ =

−2|ψ₀

_(n)|0

0|

₀|_(n) are reflection operators. By using phase estimation and thequantum Fourier transform a can be determined with accuracy O(M⁻¹).Unfortunately, the depth of the resulting quantum circuits scales asO(1/ϵ) and involves the use of a resource expensive quantum Fouriertransform. Recent developments have introduced other approaches that aimto reduce the resources to perform amplitude estimation and can removequantum phase estimation.

The most efficient variant of amplitude estimation known to date isIterative Quantum Amplitude Estimation (IQAE). It has been shownempirically that IQAE outperforms the other known variants. Although itomits quantum phase estimation, it achieves a four times betterperformance than the canonical phase estimation approach. Further, ithas been shown that for practical considerations, the following boundholds:

$\begin{matrix}{{N_{oracle}^{wc} \leq {\frac{1.4}{\epsilon}{\log\left( {\frac{2}{\alpha}{\log_{2}\left( \frac{\pi}{4\epsilon} \right)}} \right)}}},} & (19)\end{matrix}$

where N_(oracle) ^(wc) denotes the worst-case number of oracle calls,that is, applications of Q, to achieve an estimation error of ϵ>0 withconfidence level 1−, αϵ(0,1).

2.2 Path Distribution Loading

In order for Algorithm 2.1 to achieve a practical quantum advantage, theresources to perform path loading and payoff calculation are taken intoaccount. In some cases, there is an analytic form that can simplify pathloading. For example, in the case of path-independent derivatives, adistribution over paths is not involved. All that is involved is adistribution over final underlying prices S^(T), such as the log-normaldistribution given by the Black-Scholes model. This means that thedistribution could be analytically computed and then loaded eithervariationally or explicitly into quantum states. Unfortunately forquantum advantage, the analytic form for this distribution means thatthese derivatives are typically easy classically. Thus, in accordancewith various embodiments of the subject disclosure, quantum resourceestimation system 102 can focus on path dependent derivatives where asuperposition over paths is to be computed.

While the loading of general distributions is exponentially hard,several methods have been proposed. If the distribution is efficientlyintegrable, then there does exist an efficient quantum algorithm forloading, the Grover-Rudolph method. However, the algorithm has limitedapplicability in practice for derivative pricing, because the relevantprobability distributions still involve Monte Carlo integration, albeitquantumly, which is precisely what can be avoided by using AmplitudeEstimation. More details on the insufficiency of this method aredetailed in Section 7.0.

An alternative method to load the path distribution can involve use of aquantum Generative Adversarial Network (qGAN). While this has appeal forlower overhead loading, it is not yet clear how to anticipate what theoverhead from training a given qGAN will be in practice.

2.3 Error Analysis

This section investigates the various elements that contribute to theoverall error in the quantum approach to option pricing. There are threemain components that introduce error in the approach of Algorithm 2.1.Let f_(δ)=f_(max)−f_(min).

Truncation Error: The price of a derivative is determined by an integralover all the possible values of the underlying price or return. It isnot feasibly compute an integral over an infinite domain, and thusquantum resource estimation system 102 can restrict the domain ofintegration as follows: the prices and/or log-returns are restricted toa range [B_(l), B_(u)]. This restriction of the domain leaves out aprobability mass of α. Given an upper bound of P_(max) on the densityfunctions at each step and an upper bound f_(δ) on the payoff, atruncation error can be incurred which is denoted by ϵ_(truc)=P_(max)^(T)f_(δ)α.

Discretization Error: This error (denoted by ϵ_(disc)) arises from theuse of a Riemann Sum over a finite grid of points to approximate theintegral. This error can be reduced by increasing the number of qubits(n) to approximate the sum.

Amplitude Estimation Error: Amplitude estimation incurs an error ofϵ_(amp) when using 1/ϵ_(amp) repetitions of the state preparationprocedure and price computation.

The truncation and discretization errors are described in more detailbelow.

2.3.1 Truncation Error

The section presents the truncation error in return space as it thenextends to price space straightforwardly. Denote the maximum eigenvalueof the covariance matrix Σ by σ_(max). Using Chernoff tail bounds onGaussians, the probability that the log-returns for asset i lie outsideof the interval [μ_(i)−wσ_(max),μ_(i)+wσ_(max)] is upper bounded by2e^(−w) ² ^(/2). By the union bound the probability that any log-return(e.g., d assets over T time steps) lies outside the interval[B_(l)=(r−0.5σ_(max) ²)Δt−wσ_(max), B_(u)=(r−0.5σ_(max) ²)Δt+wσ_(max)]is upper bounded by 2dTe^(−w) ² ^(/2). Let the initial asset prices liein the range S_(min) ⁰,S_(max) ⁰. Then the corresponding interval inprice space is given by [S_(min) ⁰e^(B) ^(l) ^(T), S_(max) ⁰e^(B) ^(u)^(T)].

Quantum resource estimation system 102 can then define the truncatedwindow of values for dT different n-qubit registers that are w standarddeviations around the mean for each time step. The truncation error isthen given by

ϵ_(trunc)≤2dTf _(δ) e ^(−w) ² ^(/2).  (20)

2.3.2 Discretization Error

The final output of the amplitude estimation algorithm represents aRiemann Sum that approximates the truncated multidimensional integral.The integral is over dT variables corresponding to d assets over T timesteps. Assume that each underlying asset and/or return is restricted toan interval [B_(l), B_(u)]. To compute the discretization error, apply amultidimensional variation of the midpoint rule as follows: let there ben qubits used to represent each underlying asset, the domain is dividedinto 2^(ndT) cells, and corresponding to each value of the registerassociate the value of the integrand at the midpoint of thecorresponding cell. Assume that β provides an upper bound on the secondderivatives of the integrand (e.g., this can be restated as saying thatthe deviation from linearity over a range of length l is bounded byβl²/2).

Consider the error from discretization that accumulates over a singlecell. Each cell has side length (B_(u)−B_(l))/2^(n) and is a hypercubeof dimension l. Note by symmetry that the linear component of thedeviation from the value at the center of the cell integrates to 0 overthe cell. The error in each cell can thus be bounded by the integral ofthe term βx²/2 over a dT-hypercube of side length l=(B_(u)−B_(l))/2^(n)centered at the origin.

$\begin{matrix}{{\underset{\underset{dT}{︸}}{\int_{l/2}^{l/2}\mspace{14mu}{\ldots\mspace{14mu}\int_{l/2}^{l/2}}}\beta\;{x^{2}/2}} = {{l^{{dT} - 1}{\beta\left\lbrack {\beta{x^{3}/6}} \right\rbrack}_{l/2}^{l/2}} = {\frac{\beta l^{{dT} + 2}}{24} = {\frac{{\beta\left( {B_{u} - B_{l}} \right)}^{{dT} + 2}}{24 \cdot 2^{n{({{dT} + 2})}}}.}}}} & (21)\end{matrix}$

Aggregating the error over all the cells provides

$\begin{matrix}{\epsilon_{disc} = \frac{{\beta\left( {B_{u} - B_{l}} \right)}^{{dT} + 2}}{24 \cdot 2^{2n}}} & (22)\end{matrix}$

In terms of the number of standard deviations used in the discretizationand the largest eigenvalue of the covariance matrix a the totaldiscretization error is bounded by

$\begin{matrix}{\epsilon_{disc} \leq {\frac{{\beta\left( {2W\sigma_{{ma}x}} \right)}^{{dT} + 2}}{24 \cdot 2^{2n}}.}} & (23)\end{matrix}$

For a target discretization error, Equation (23) also gives the totalnumber of qubits that can be used to represent d assets for T timesteps,given by

ndT=dT┌½(log₂(β/24)−log₂(ϵ_(disc))+(dT+2)log₂(2wσ _(max)))┐  (24)

The truncation and discretization errors apply in general to the methodsintroduced herein, though each method has additional method-specificerror sources which are discussed separately for each method.

3.0 Methods for Advantage in Quantum Derivative Pricing

The following sections describe two approaches that can work effectivelyto perform quantum derivative pricing in practice: Riemann summation andthe re-parameterization method of the subject disclosure. Riemannsummation was introduced previously and described herein in accordancewith one or more embodiments of the subject disclosure is the firstresource analysis for its application to achieve a quantum advantage.This analysis uncovers limitations in error scaling due tonormalization. The new re-parameterization method that can be definedand/or implemented by quantum resource estimation system 102 inaccordance with one or more embodiments of the subject disclosure asdescribed below avoids the downsides of other methods and offers thefirst end to end path to quantum advantage in practice.

3.1 Riemann Summation

The Riemann summation method gives an approach to construct the

path loading operator in Algorithm 2.1. Let N=2^(ndT) be the size of theHilbert space that contains all possible paths. Let {tilde over(P)}_(max) be the maximum value of the d-asset multivariate transitionprobabilities from Equation (2). Then {tilde over (P)}({right arrow over(S)}^(t)|{right arrow over (S)}^(t−1))=P({right arrow over(S)}^(t)|{right arrow over (S)}^(t−1))/{tilde over (P)}_(max)ϵ[0,1] isthe normalized transition probabilities over all choices of {right arrowover (S)}^(t) and {right arrow over (S)}^(t−1). Let the asset price foreach asset at each timestep be discretized in the interval [0, S_(max)].The steps of the method summarized in Algorithm 3.1 defined belowcalculate the price of the derivative with a normalization factor1/P_(maxf) ^(T), with P_(max)={tilde over (P)}_(max)S_(max) ^(d). Notethat the normalization factor in the final step scales exponentially inT. If P_(max)<1 no normalization is involved and this factor isredundant. However, if P_(max)>1, the error increases exponentially,which renders this approach impractical.

Algorithm 3.1—Riemann Summation Pricing

Use parameters n, d, and T that are all positive integers.

Obtain access to operators W_(t), t=1, . . . , T that apply thetransition probabilities of the stochastic process into an ancilla via

W _(t) |{right arrow over (S)} ^(t)

_(n) |{right arrow over (S)} ^(t−1)

_(n)|0

|{right arrow over (S)} ^(t)

_(n) |S ^(t−1)

_(n)[√{square root over (1−{tilde over (P)}({right arrow over (S)} ^(t)|{right arrow over (S)} ^(t−1)|0

+)}√{square root over ({tilde over (P)}({right arrow over (S)} ^(t)|{right arrow over (S)} ^(t−1))}|1

]  (25)

1. Apply Hadamards to ndT qubits to prepare an equal superposition ofall paths.

2. Load the initial prices {right arrow over (S)}⁰ into the zero-thnd-qubit register.

3. Apply each of the T transition operators W_(t) to construct

$\begin{matrix}{{{\frac{1}{\sqrt{N}}{\sum_{\omega}{\left. {{\overset{\rightarrow}{S}}^{0}\mspace{14mu}\ldots\mspace{14mu}{\overset{\rightarrow}{S}}^{T}} \right\rangle\left\lbrack \mspace{14mu}{\ldots + {\sqrt{\prod_{t = 1}^{T}{\overset{\sim}{P}\left( {{\overset{\rightarrow}{S}}^{t}❘{\overset{\rightarrow}{S}}^{t - 1}} \right)}}\left. {1\mspace{14mu}\ldots\mspace{14mu} 1} \right\rangle_{T}}} \right\rbrack}}} = {\frac{1}{\sqrt{P_{\max}^{T}}}\frac{1}{\sqrt{N}}{\sum_{\omega}{\left. {{\overset{\rightarrow}{S}}^{0}\mspace{14mu}\ldots\mspace{14mu}{\overset{\rightarrow}{S}}^{T}} \right\rangle\left\lbrack \mspace{14mu}{\ldots + {\sqrt{p(\omega)}\left. {1\mspace{14mu}\ldots\mspace{14mu} 1} \right\rangle_{T}}} \right\rbrack}}}},} & (26)\end{matrix}$

where N=2^(ndT).

4. Calculate δ(ω)=arcsin√{square root over ({tilde over (f)}(ω))} into aquantum register, obtaining

$\begin{matrix}{\frac{1}{\sqrt{P_{\max}^{T}}}\frac{1}{\sqrt{N}}{\sum_{\omega}{{\left. {{\overset{\rightarrow}{S}}^{0}\mspace{14mu}\ldots\mspace{14mu}{\overset{\rightarrow}{S}}^{T}} \right\rangle\left\lbrack \mspace{14mu}{\ldots + {\sqrt{p(\omega)}\left. {1\mspace{14mu}\ldots\mspace{14mu} 1} \right\rangle_{T}}} \right\rbrack}{\left. {\delta(\omega)} \right\rangle.}}}} & (27)\end{matrix}$

5. Introduce an ancilla qubit and rotate the value of the {tilde over(f)}(ω) register into its amplitude:

$\begin{matrix}{\ldots + {\frac{1}{\sqrt{P_{\max}^{T}}}\frac{1}{\sqrt{N}}{\sum_{\omega}{\sqrt{{p(\omega)}{\overset{\sim}{f}(\omega)}}\left. \omega \right\rangle\left. {1\mspace{14mu}\ldots\mspace{14mu} 1} \right\rangle_{T}{\left. 1 \right\rangle.}}}}} & (28)\end{matrix}$

6. Use amplitude estimation to extract the probability of the ancillabeing |1

, which is the (e.g., discretized) expected payoff

({tilde over (f)}(ω)/P_(max) ^(T)). Rescale this to obtain

(f)=P_(max) ^(T)((f_(max)−f_(min))

({tilde over (f)})+f_(min)).

The normalization factor P_(max) is easier to handle in return spacewhere the probability density function is given by Equation (8). If thelog-returns are discretized at each timestep for each asset to ±w timesthe asset's volatility σ_(j), then

$\begin{matrix}{{P_{\max} = \frac{\left( {2W} \right)^{d}{\prod_{j = 1}^{d}\sigma_{j}}}{\left( {2\pi} \right)^{d/2}\left( {\det\sum} \right)^{1/2}}}.} & (29)\end{matrix}$

When the d assets are uncorrelated, then

$\begin{matrix}{{P_{\max} = \left( \frac{2W}{\sqrt{2\pi}} \right)^{d}},} & (30)\end{matrix}$

and therefore, choose w≤π/√{square root over (2)}, for P_(max)≤1.However, choosing a small discretization window w increases thetruncation error described in Section 2.3.1, and for w≤π/√{square rootover (2)} then ϵ_(trunc)≥2e^(−π) ² ^(/4)˜0.17, which increasesproportionally to the number of assets and timesteps in the computation.

3.1.1 Riemann Summation Error Analysis

In addition to the truncation and discretization errors from Section2.3, the Riemann summation approach includes errors due to scalingconsiderations and quantum arithmetic.

When working in return space, one transition operator is used whichcomputes Equation (12) and performs the amplitude encoding of √{squareroot over (p(ω))} in Equation (26). Assuming the transition operatorintroduces a maximum additive error ϵ_(dens) and the payoff operatorcomputing Equation (27) and Equation (28) introduces payoff error ϵ_(f),the total arithmetic error of the quantity that can be estimated usingamplitude estimation is

$\begin{matrix}{\epsilon_{arith} = {\frac{1}{N}{\sum_{\omega}{\left\lbrack {{\left( {{p(\omega)} + \epsilon_{dens}} \right)\left( {{f(\omega)} + \epsilon_{f}} \right)} - {{p(\omega)}{f(\omega)}}} \right\rbrack.}}}} & (31)\end{matrix}$

Ignoring quadratic error terms, then

$\begin{matrix}{{\epsilon_{arith} \approx {{\frac{1}{N}{\sum_{\omega}{{p(\omega)}\epsilon_{f}}}} + {\frac{1}{N}{\sum_{\omega}{{f(\omega)}\epsilon_{dens}}}}} \leq {\frac{\epsilon_{f}}{\left( {2W\sigma_{\max}} \right)^{dT}} + {f_{\delta}\epsilon_{dens}}}},} & (32)\end{matrix}$

where the log-returns for each asset and each timestep have beenconstructed to discretize the domain [−wσ_(max), wσ_(max)].

The probability density error ϵ_(dens) arises from the computation of|arcsin √{square root over (P({right arrow over (R)}))}

with P({right arrow over (R)}) given by Equation (12), and the ancillarotation in Equation (26). The term inside the exponential in Equation(12) can be written as

−½Σ_(t=1) ^(T)({right arrow over (R)} ^(t)−{right arrow over(μ)})^(T)Σ⁻¹({right arrow over (R)} ^(t)−{right arrow over(μ)})=−½Σ_(t=1) ^(T)Σ_(i=1) ^(d) Σj=i ^(d) C _(ij) R _(i) ^(t) R _(j)^(t),  (33)

where R=R−μ and C_(ij) are classical variables containing volatility andcorrelation parameters from the correlation matrix Σ. In Equation (33),each calculation of R thus incurs an error of ϵ_(A) and there are(d+d₂)·T multiplications in total. Each R term is bounded by |w|σ_(max)by construction, where each quantum register representing a log-return Ris constructed to represent values in the window [−wσ_(max), wσ_(max)].Using the error analysis for addition and multiplication in Section 8.2,the total error in computing Equation (33) is

$\begin{matrix}{\epsilon_{sum} = {\left( {\frac{{2w\sigma_{\max}} + n}{2^{n - p}} + \frac{1}{4^{n - p}}} \right){\left( {d + d_{2}} \right) \cdot {T.}}}} & (34)\end{matrix}$

Then using the error propagation analysis in Section 8.2 to compute theexponential, square root, arcsine and sine functions on quantumregisters which already contain arithmetic errors, ϵ_(dens) can bebounded by

ϵ_(dens)≤ϵ_(sin)+ϵ_(arcsin)−arcsin(0.5−(ϵ_(sq)+√{square root over(ϵ_(exp)+ϵ_(sum))}))+arcsin(0.5)  (35)

Each rescaling performed to the input variables introduces acorresponding rescaling error. In addition to the P_(max) rescalingdescribed in the previous section, quantum resource estimation system102 can also scale the payoff by 1/f_(δ) to lie in [0,1]. The finalanswer can be multiplied by P_(max) ^(T)f_(δ) a to account for theserescalings, and the error in the estimate of the truncated integral byamplitude estimation is therefore scaled by P_(max) ^(T)f_(δ). Quantumresource estimation system 102 can then bound the error in the RiemannSummation approach

ϵ_(total)≤ϵ_(trunc) +P _(max) ^(T) f_(δ)(ϵ_(disc)+ϵ_(arith)+ϵ_(amp)),  (36)

where ϵ_(trunc), ϵ_(disc), and ϵ_(amp) are defined as in Section 2.3.

3.1.2 Resource Estimates

As an example, consider a basket auto-callable with 5 autocall dates andparameters T=20, d=3, and target an error of ϵ_(total)/f_(δ)≤2×10⁻³.Choose w˜5 for the truncation error in Equation (20) to be within thetotal target error, and Equation (30) gives P_(max)≈4³. This makes thescaling factor prohibitively large with P_(max) ^(T)≈10⁴⁰. However,there may be some methods to deal with this normalization issue, such asa method inspired by importance sampling and discussed in Section 9.2.Thus, assume that some method is invented to deal with thenormalization, and set P_(max)=1.

Then, using resource calculations as discussed in Section 9.1, boundϵ_(arith)≤2×10⁻³ with n=34 and p=2. Here p is the integer part of thefixed point representation as defined in Equation (63). The Q operatorin this case uses 21 k qubits and has a T-depth of 23 k, including theresources to compute prices from log-returns using Equation (11). For achoice of Δt=1/20 and 0 min=0.1 compute that β≈17. Choose σ_(max)=0.4and w=5. Thus for the choice of n, ϵ_(disc)≈f_(δ)10⁻⁵ and

ϵ_(trunc) ≤f _(δ)·10⁻⁴.  (37)

If a target ϵ_(amp) is chosen for the amplitude estimation of 10⁻³ and atarget confidence level of α=10⁻² then N_(oracle) ^(wc)≤8 k can beobtained. This means that the total T-depth is about 1.9×10⁸.

Using the same analysis, for a TARF contract (e.g., with reference toSection 4.2) with d=1, T=26 and Δt=1/26, assuming the underlying hasannualized volatility σ=0.4, a target error of ϵ_(total)/f_(δ)≤2×10⁻³can be achieved with a total T-depth of 1.7×10⁸ and 15 k qubits.

3.2 Re-Parameterization Method

The limitations of normalization in Riemann summation motivate thedevelopment and implementation of a new method to load stochasticprocesses. In the re-parameterization method, quantum resourceestimation system 102 can shift to modelling assets in return space. Asdescribed in Section 1.2, in return space underlying assets includeuncorrelated normal distributions. Recognize that these differentdistributions can be loaded by preparing (e.g., in parallel) manystandard normals and then applying affine transformations to obtain themeans and standard deviations. This approach extracts a specificsubroutine-loading a standard normal into a quantum state—and uses it asa resource to load the full distribution of underlying paths. The normalloading subroutine itself can then be precomputed and optimized usingvariational methods. This is an advantageous combination offault-tolerant quantum computing with variational compilation and willbe discussed in Section 3.2.1. Overall the re-parameterization methodavoids the normalization issues in Riemann summation and reduces thecomputational costs and/or resources. The steps in re-parameterizationpricing are described in Algorithm 3.2 defined below. Note that a pathω_(R)ϵΩ_(R) in this context refers to a series of log-returns {rightarrow over (R)}¹, . . . , {right arrow over (R)}^(T). There-parameterization method removes the problematic dependence onP_(max), and the operators G_(j) can be implemented with relatively fewresources using variationally trained circuits as described in thefollowing sections.

Algorithm 3.2—Re-Parameterization Method Pricing

Use parameters n, d, and T that are all positive integers.

Obtain access to an operator

that loads a standard Gaussian distribution Σ_(i)√{square root over(g_(i))}|i

into an n-qubit register. Let g_(i) be the value of the probability massfunction for a standard Gaussian distribution discretized into2^(n)-bins.

1. Apply dT Gaussian operators

, to ndT qubits. This constructs

$\begin{matrix}{{{\underset{t = 1}{\overset{T}{\otimes}}{\underset{j = 1}{\overset{a}{\otimes}}{\mathcal{G}\left. 0 \right\rangle_{n}}}} = {\sum_{\omega_{\overset{\_}{R}}}{\sqrt{p\left( \omega_{\overset{\_}{R}} \right)}\left. \omega_{\overset{\_}{R}} \right\rangle_{ndT}}}},} & (38)\end{matrix}$

where ω _(R) runs over all 2^(ndT) different realizations of thismultivariate standard Gaussian, and p(ω _(R) ) denotes the correspondingprobabilities.

2. Let Σ=LL^(T) be the Cholesky decomposition of the covariance matrix.Perform affine transformations {right arrow over (R)}^(t)={right arrowover (μ)}^(t)+{right arrow over (L)} ^(T)R^(t) to adjust the center andvolatility of each Gaussian. Denote the corresponding return paths andprobabilities by ω_(R) and p(ω_(R)), respectively.

3. If the payoff can be computed directly from the log-returns, thendirectly calculate δ(ω_(R))=arcsin√{square root over ({tilde over(f)}(ω_(R)))} into a quantum register

Σ_(ω) _(R) √{square root over (p(ω_(R)))}|ω_(R)

|δ(ω_(R))

.  (39)

If the payoff is defined in terms of prices and not just log-returns,then first compute the price space path ω for each asset using {rightarrow over (S)}^(t)={right arrow over (S)}⁰e^(Σ) ^(j=1) ^(t)^({right arrow over (R)}) ^(j) . This calculation can be done inparallel for each asset (e.g., each derivative asset).

4. Introduce an ancilla qubit and rotate the value of the f(ω_(R))register into its amplitude:

Σ_(ω) _(R) √{square root over (p(ω_(R))(1−{tilde over(f)}(ω_(R))))}|ω_(R)

|0

+Σ_(ω) _(R) √{square root over (p(ω_(R)){tilde over (f)}(ω_(R))))}|ω_(R)

|1

.  (39)

5. Use amplitude estimation to extract the probability of the ancillabeing |1

, which is the (e.g., discretized) expected payoff

({tilde over (f)}). Rescale this to obtain

(f)=(f_(max)−f_(min))

({tilde over (f)})+f_(min).

3.2.1 Variationally Trained Gaussian Loaders

The standard Gaussian loading operator G can be pre-computed because, inthe re-parameterization method, it is problem independent. This sectiondescribes an approach to variationally optimize this operator. Considerthe case of preparing a standard normal distribution g(x_(i)) defined adiscretized mesh of points x_(i)=−w+iΔx, with i=0, . . . 2^(n)−1, andΔx=2w/2^(n). In the following example the domain is fixed to w=5 so thatthe full range of value considered is 2w=10. This choice leaves outsidethe domain a probability mass of ˜5×10⁻⁷. Finally, take into account thedifferent metrics used to normalize a function in real-space and awafefunction in a quantum register, which is normalized in such a waythat the sum of its squared elements is one, therefore, aim to load inthe quantum register the following distribution (e.g., the targetdistribution):

g(x _(i))×Δx,Σ _(i) g(x _(i))×Δx=1.  (41)

Notice that, due to the finite truncation domain, the targetdistribution is normalized to 1−α. In principle, the distribution can bere-normalized to one in the chosen interval of width 2w. Either way thischoice provides a negligible difference when compared with the errorobserved in the training.

The variational ansatz of choice is represented by a so-calledRy−Controlled NOT (Ry−CNOT) ansatz, with linear connectivity (e.g., withreference to Section 11.0). The various embodiments of the subjectdisclosure provide a novel strategy to optimize the circuit in thiscontext, which relies on a energy-based method, and is also detailed inSection 11.0. In short, the target cost function is the energy of theassociate quantum harmonic oscillator problem, whose ground state isnaturally Gaussian. Note that the solution (e.g., the modulus square ofthe solution) of the discretized quantum harmonic oscillator cancoincide to a Normal distribution only in the limit of Δx→0. To fixthis, perform a subsequent re-optimization targeting directly theinfinity-norm between the two distributions.

$\begin{matrix}{L_{\infty} = {\max\limits_{i}{{{{g\left( x_{i} \right)} \times \Delta\; x} - {\overset{\sim}{g}\left( x_{i} \right)}}}}} & (42)\end{matrix}$

where the quantum state encoded in the register is defined bycoefficients √{square root over ({tilde over (g)}(x_(i)))}.

Notice that training directly with Equation (42) as a cost function isnot efficient and a pre-training using the energy based approach isused. Observe how the L_(∞) cost-function displays a much morecorrugated landscape in the circuit parameter space compared to theenergy of the associate quantum harmonic osclillator problem.

Notice that the circuits to encode these Gaussian states for differentchoices of register size n can be pre-trained in advance and so trainingis not included as part of the runtime for any given derivative pricingproblem. Show in FIG. 4 are results for different register sizes anddepth of circuit ansatz. More details are provided in Section 11.0.

FIG. 4 illustrates an example, non-limiting graph 400 that canfacilitate estimation of quantum resources to calculate an expectationvalue of a stochastic process using a re-parameterization method inaccordance with one or more embodiments described herein. Repetitivedescription of like elements and/or processes employed in respectiveembodiments is omitted for sake of brevity.

Graph 400 illustrates L_(∞) errors from training (e.g., via quantumresource estimation system 102 and/or variational component 202)variational Ry-CNOT circuits to approximate G for different registersizes n. This numerical study shows that the state that can be preparedvariationally approaches the target exponentially fast in the depth,hence in the number of gate operations. This observation is in goodagreement with the expected behaviour from the Solovay-Kitaev theorem,that provides an upper bound for the number of gates that can be used toachieve a desired accuracy for cost function. Indeed, for any targetoperation UϵSU(2^(n)), there is a sequence S=U_(s) ₁ U_(s) ₂ . . . U_(s)_(D) of operators in a dense subset of SU(2^(n)), such that error in theenergy ξ decreases exponentially with the depth D=O(log^(c)(1/ε)).Although the subset of SU(2^(n)) operations generated by the entanglerblocks in the circuit does not generate a dense subset of SU(2^(n))arbitrarily close to the exact unitary U (e.g., generator of the targetstate), it can be numerically observed that the exponential decrease ofthe error with the number of gates still holds.

This section ends by investigating the portability of these results inthe fault-tolerant regime, which can enable application of the wholederivative pricing algorithm. While the numerical results provideevidence for a rather efficient Gaussian state preparation in terms ofcircuit depths for a Ry−CNOT, an additional step is made in view of afault-tolerant implementations of such circuit. In this new-framework,the continuous rotation Ry gate is expanded as a finite product ofdiscrete operations. Following again the Solovay-Kitaev theorem, or morespecialized results [27], it is possible to have also an efficientrepresentation of any SU(2) operator with a sequence of Clifford+T-gatesthat scale logarithmically with the threshold error ϵ. Investigate howthe results obtained before can be transferred in this regime whererotation's angles can take only discretized value. Therefore, assumethat each parameter ϑ_(q) _(j) ^(k) can only be represented by theformat j*2π/M_(digit), where j is an integer. Numerically show inSection 11.0 is how the error introduced by such digitization decreasessystematically with the mesh size as 1/M_(digit).

3.2.2 Error Analysis

The total error in the re-parameterization approach is

$\begin{matrix}{{\frac{\epsilon_{t{otal}}}{f_{\delta}} \leq {\epsilon_{trunc} + \epsilon_{disc} + \epsilon_{arith} + \epsilon_{amp}}},} & (43)\end{matrix}$

where ξ_(trunc), ϵ_(disc), and ϵ_(amp) are the truncation,discretization, and amplitude estimation error bounded in Section 2.3.Here, the term ϵ_(arith) arises from the individual errors introducedduring the preparation of the Gaussians and the calculation of thepayoff. Assuming that each Gaussian g(x_(i)) prepared has L_(∞) errorϵ_(dens) and the payoff calculation introduces a max error of ϵ_(f), thetotal error will be

$\begin{matrix}{{\epsilon_{arith} = {\underset{\underset{dT}{︸}}{\sum_{x_{1} = {- w}}^{w}\mspace{14mu}{\ldots\mspace{14mu}\sum_{x_{dT} = {- w}}^{w}}}\left\lbrack {{\prod_{i = 1}^{dT}{\left( {{g\left( x_{i} \right)} + \epsilon_{dens}} \right)\left( {{f(x)} + \epsilon_{f}} \right)}} - {\prod_{i = 1}^{dT}{{g\left( x_{i} \right)}{f(x)}}}} \right\rbrack}},} & (44)\end{matrix}$

where x=(x₁, x₂, . . . , x_(dT)). Expanding the integrand and keepingonly the linear error terms, gives

ϵ_(arith)≤2wdTf_(δ)ϵ_(dens)+ϵ_(f),  (45)

where Σ_(−w) ^(w)g(x)≤1 is used due to truncation of the probabilitymass function.

3.2.3 Resource Estimates

Calculate the resources that can be involved for the same basketauto-callable as in Section 3.1.2, where d=3, T=20, Δt=1/20,σ_(max)=0.4, σ_(min)=0.1 and w=5, and the contract has 5 autocall dates.Further assume that each Gaussian is prepared with n=5 qubits, such thatϵ_(dens)=2×10⁻⁶, ϵ_(amp)=ϵ_(f)=10⁻⁴, which gives a total error ofϵ_(total)/f_(δ)≈2×10⁻³. From FIG. 4 it can be observed that Gaussianstates can be prepared with L_(∞)˜2×10⁻⁶ using 5 qubits and circuitdepth 6, requiring 7 layers of Ry gates. With these inputs and using theresource calculations described in Section 10.0, constructing the Qoperator using re-parameterization involves 7.5 k qubits and has aT-depth of 5.7 k, which includes the computation of prices fromlog-returns, Equation (11). For a target confidence level of α=10⁻², thetotal T-depth is 4.6×10⁷. With the re-parameterization method, pricingthe TARF of Section 3.1.2 with d=1, T=26, Δt=1/26 and σ=0.4 to accuracyϵ_(total)/f_(δ)≈2×10⁻³ uses a total T-depth of 6×10⁷ and 9.5 k qubits.

4.0 Payoffs

4.1 Auto-Callable Contracts

An auto-callable contract is typically defined in terms of asset returnsrelative to predefined reference levels, and includes a notional valuewhich is used to calculate the dollar value of the contract. For asingle underlying, an auto-callable can include:

-   -   a set of binary options {(K_(i), t_(i), f_(i))}_(i=0 . . . m−1)        each with strike K_(i), exercise time t_(i), and fixed payoff        f_(i). Assume these are sorted so that t_(i)<t_(i+1).    -   a short knock-in put with strike K_(o) and barrier b, and    -   the condition that if any binary option pays off then all        subsequent options at later times and the put are knocked out.

The strike and barrier parameters are defined in terms of returns of theunderlying asset price S(t) relative to a reference level, which withoutloss of generality can be taken to be the initial spot price of theunderlying S₀. The payoff f_(i) of the binary options is similarlydefined as a dimensionless parameter denoting a return. In return space,where the basis vectors represent log-returns of the underlying asset(see Section 1.2), checking whether the underlying asset has crossed astrike or barrier K given a log-return value R, can involve checkingwhether e^(R)≥K. Let {tilde over (f)}_(i) be the normalized payoff givenby

$\begin{matrix}{{\overset{\sim}{f}}_{i} = {\frac{{e^{- {rc}_{i}}f_{i}} + {e^{- {rt}_{m - 1}}K_{o}}}{e^{- {rt}_{\max_{f_{\max} + e^{- {rt}_{m - 1}}}}}K_{o}}.}} & (46)\end{matrix}$

Algorithm 4.1—Auto-Callable Payoff Implementation

Obtain an auto-callable with parameters {(K_(i), t_(i),f_(i))}_(i=0 . . . m−1), K_(o) and b.

1. For each time step t=1 . . . T assume access to the cumulative return|e^(Σ) ^(i=0) ^(t) ^(R) ^(t)

=|R_(c) ^(t)

, calculated by the path distribution loading method of choice.

2. For each t_(i) apply in parallel a comparator to obtain the strikeregister |R_(c) ^(t) ^(i) <K_(i)

_(m).

3. Let θ_(i)=arcsin(√{square root over ({tilde over (f)}_(i))}).Serially, for each bit of the strike register apply a controlledrotation of θ_(i) into an accumulator qubit conditioned on all previousbits having been zero. This is illustrated in FIG. 5. This introduces anm-qubit ancilla register a.

4. For each cumulative return |R_(c) ^(t)

apply in parallel a comparators to obtain a register |R_(c)^(t)<bANDR_(c) ^(t)<K_(o)

_(T), denoting if the put option has been knocked in and if it is in themoney. Then OR these bits together to obtain |put

₁, which holds whether the payoff from the put option is considered.

5. Compute |R_(c) ^(t)−K_(o)

and normalize it using Equation (46) to obtain the put option payoff|f_(p)

. Compute arcsin(√{square root over (f_(p))})

.

6. Then control on both |put

and a[m−1] to rotate |arcsin(√{square root over (f_(p))})

bitwise into the target qubit using controlled R_(y) rotations.

FIG. 5 illustrates an example, non-limiting circuit 500 that canfacilitate estimation of quantum resources to calculate an expectationvalue of a stochastic process using a re-parameterization method inaccordance with one or more embodiments described herein. Repetitivedescription of like elements and/or processes employed in respectiveembodiments is omitted for sake of brevity.

Circuit 500 comprises an example, non-limiting circuit that can be usedto accumulate binary option payoffs in an auto-callable with 5 binaryoptions. Here the qubits s₀, . . . , s₅ represent the booleancomparisons for the 5 strike values K_(i). A payoff f_(i) (e.g., givenby a particular phase in the RY rotation) only occurs if no previouspayoff has happened. The overall payoff is loaded into the amplitude ofqubit e₀.

Amplitude estimation allows calculation of the expected return of thecontract

_(R)(f), and its dollar value will be given by V=N·

_(R)(f), where N is the notional value specified in the contract. Anauto-callable can also be defined on a basket of assets instead of justone. Typical examples include BestOf and WorstOf, where the return ofthe contract is based on the return of the best or the worst performingasset in the basket respectively. These can be treated similarly to thesingle-asset case where the return of each asset |R_(c) ^(T)−K_(o)

_(j=1 . . . d) is first compared to find the largest or smallest (e.g.,as appropriate).

Steps 2 and 4 in Algorithm 4.1 defined above can be performed withlogical operation circuits (e.g., Comparator, AND, OR) which introduceno error, while steps 3 and 6 involve use of controlled-Ry rotationswhose decomposition into T-gates is a function of an additive error E,which can be chosen depending on the desired accuracy of thecalculation. Step 5 is the most resource heavy component of the payoffcircuit, which can involve the computation of the quantum register|R_(c) ^(t)−K_(o)

, the division of that register by the classical constant in thedenominator of Equation (46), as well as the computation of the squareroot and arcsine of the register. Describe in detail in Section 8.1 arethe resource criteria for all the above circuit components, and thecorresponding arithmetic and gate synthesis error in Section 8.2.

Again consider the auto-callable contract from Section 3.1.2 and Section3.2.3 with 5 autocall dates, defined on d=3 assets and simulated usingT=32 timesteps. Target a total additive payoff error ϵ_(f) which whendistributed across the operations of steps 3, 5, 6 in Algorithm 4.1determines the resources that can be used by each component. Forϵ_(f)=10⁻⁴, the circuit computing the auto-callable payoff involves 1.6k qubits and a T-depth of 2 k, assuming computations can be parallelizedwherever possible.

4.2 TARFs

This section describes the TARF implementation for a single underlyingin price space.

A TARF is:

-   -   A forward price F, payment dates t₁, . . . , t_(m), and two        strike prices K_(upper) and K_(lower)≤F, a knock-out price        K_(o), and an accrual cap Cϵ        ₊. Assume dates are sorted so that t_(i)≤t_(i+1).    -   At each time t_(i) the TARF has a payoff

$\begin{matrix}{f_{i} = \left\{ \begin{matrix}{{S^{i} - {FifS}^{i}} > K_{upper}} & \; \\{{2\ \left( {S^{i} - F} \right){ifS}^{i}} < K_{{lowe}r}} & \; \\0 & {{otherwise}.}\end{matrix} \right.} & (47)\end{matrix}$

-   -   a knock-out condition that if at any t_(i) the price is above        K_(o)>F all subsequent payoffs are knocked out    -   an accrual cap condition such that if the total gain accumulated        by any payment date exceeds C the contract holder only receives        a payoff such that the total gains equals C and the rest of the        forward contracts are knocked out.

Let {tilde over (f)}_(i) be the normalized payoff given by

$\begin{matrix}{{\overset{\sim}{f}}_{i} = {\frac{{e^{- {rt}_{i}}f_{i}} + {\sum_{j = 1}^{m - 1}{e^{- {rt}_{j}}2TF}}}{{\sum_{j = 1}^{C/{({K_{o} - F})}}{e^{- {rt}_{j}}\left( {K_{o} - F} \right)}} + {\sum_{t_{j} = 1}^{m - 1}{e^{- {rt}_{j}}2TF}}}.}} & (48)\end{matrix}$

Algorithm 4.2—TARF Payoff Implementation

Obtain a TARF with parameters (F, t₁, . . . , t_(m), K_(upper),K_(lower), K_(o), C).

1. Begin with a knock-out and accrual cap qubit |0

_(o), |0

_(c)

2. For all times t_(i) apply in parallel three comparators and combinesome of their results to obtain the registers |S^(t) ^(i) >K_(o)

_(m), |K_(upper)<S^(t) ^(i) <K_(o)

_(m), |S^(t) ^(i) <K_(lower)

_(m)

3. For each t_(i):

-   -   (a) OR |S^(t) ^(i) >K_(o)        _(i) with |⋅        _(o) onto a new qubit which can be relabelled |⋅        _(o)    -   (b) Compute the payoff (e.g., conditioned on the different        strike qubits) and add the payoff to a register keeping track of        the total payoff to obtain |Σ_(j=1) ^(i)f_(j)        _(i)    -   (c) Apply a comparator to compute whether the accrual cap has        been reached to obtain |Σ_(j=1) ^(i)f_(j)>C        _(i)    -   (d) Compute the amount that makes the total equal to C to obtain        |C−Σ_(i=1) ^(i−1)f^(j)        _(i)    -   (e) Compute the normalized payoff {tilde over (f)}_(i) in the        cases for which the accrual cap was and was not reached in        parallel and add the appropriate one controlled on the AND of        the respective condition, the NOT of the (e.g., new) knock out        qubit |⋅        _(o) and NOT of the accrual cap qubit |⋅        _(c) to obtain |Σ_(j=1) ^(i){tilde over (f)}_(j)        _(i).    -   (f) AND the |Σ_(i=1) ^(i−1)=f_(j)>C        _(i) and cap qubit |⋅        _(c) and store the result into a new qubit which can be        relabelled |⋅        _(c)

4. Compute θ=arcsin(√{square root over (Σ_(j=1) ^(T){tilde over(f)}_(j))}) into |θ

5. On a final encoding qubit |0

_(e), apply a series of controlled R_(y) rotation of angle 2^(−i)controlled for each qubit i of |θ

.

5.0 Discussion

The various embodiments of the subject disclosure described hereinprovide a thorough resource and error analysis to price financialderivatives using quantum computers. In particular, these variousembodiments use auto-callables and TARFs as example case studies, whichare two types of path-dependent options that are relevant in practiceand difficult to price classically. To this extent, the variousembodiments of the subject disclsoure provide a new method to loadstochastic processes that overcomes the limitations of existingapproaches. Although these various embodiments involve geometricBrownian motion, the subject disclosure is not so limiting, as theapproach described herein can be easily extended, for example, tostochastic or local volatility methods by loading multiple independentstochastic processes and having a conditional or non-stationaryre-parametrization.

The resource estimates provided herein give a target performancethreshold for quantum computers capable of demonstrating advantage inderivative pricing. Assuming a target of 1 second for pricing anauto-callable option, the quantum processor would have a logical clockrate of 10 MHz at a code distance that can support 10¹ 1 logicaloperations.

Although current estimates of the logical clock rate are around 10kilohertz (kHz), that is, approximately three orders of magnitude below,future work on algorithms, circuit optimization, error correction, andhardware will continue to improve the resource criteria and runtimes.For example, in the case of Shor's algorithm, the estimated resourcecriteria have reduced by almost three orders of magnitude throughcareful analysis across several publications. The subject disclsouredescribed herein represents the first milestone on the journey towardsQuantum Advantage for pricing financial derivatives.

6.0 Background on Derivatives

6.1 Forwards

An example of a derivative is a forward contract, often simply called aforward. With a forward, the holder promises to buy or sell a certainasset to the issuer on a specified date in the future at a fixed price Fknown as the forward price. A simple path-independent example is wherethe holder promises to buy x amount of an asset at F dollars per asset mmonths from now. Forwards are typically settled in cash, that is,instead of the money and asset exchanging hands on the expiration date,a payoff is determined based on the value of the asset and there is onlyan exchange of money determined by this payoff. For example, if theprice at the expiration date T of the asset is S^(T), the payoff isgiven by f(S^(T))=x(S^(T)−F), where if f(S^(T))>0, the contract holdermakes a profit (and the issuer a loss) and the opposite if f(S^(T))<0.

6.2 Options

Another example of a derivative is an option. Options can be viewed asconditional forwards. With an option contract, the holder has the optionto buy or sell a certain asset to the issuer on some future date at apre-determined price, unlike the forward where the issuer is obliged tobuy or sell the asset. If the holder chooses to buy or sell the asset,they have chosen to exercised the option. Similarly to the forwards,option contracts are usually settled in cash based on the value of theasset on the exercise date. An example of a path-independent option witha single underlying asset is a European call option, where the issuerhas the option of buying an asset at a strike price K on expirationdate. The payoff on expiration date can then be written asf(S^(T))=max(S^(T)−K,0). A European put option is where the issuer hasthe option of selling an asset at a strike price K on expiration date,which gives a payoff of f(S^(T))=max(K−S^(T),0). Another example of apath-independent option is a binary option which has a fixed payoff ifthe underlying asset is above or below the strike at time T.

6.3 Path-Dependence and Discounted Payoffs

An example of a path-dependent derivative is a knock-out European calloption. This is the same as a European call option, but with anadditional knock-out price π. If at any time from 0 to T the underlyingasset goes above this value, then the contract is worth nothing. Thispath-dependent payoff function has the form

$\begin{matrix}{{f\left( {S^{0},S^{1},\ldots\mspace{14mu},S^{T}} \right)} = \left\{ \begin{matrix}{S^{T} - K} & {{{{if}\mspace{14mu} S^{T}} > {K\mspace{14mu}{and}\mspace{14mu} S^{i}} < \pi},{\forall{i \in \left\{ {0,\ldots\mspace{14mu},T} \right\}}}} \\0 & {{otherwise}.}\end{matrix} \right.} & (49)\end{matrix}$

The inclusion of the value of the underlying at times other than T iswhat introduces path dependence. Another example is a knock-in t optionwhich has payoff

$\begin{matrix}{{f\left( {S^{0},S^{1},\ldots\mspace{14mu},S^{T}} \right)} = \left\{ \begin{matrix}{K - S^{T}} & {{{{if}\mspace{14mu} S^{T}} > {K\mspace{14mu}{and}\mspace{14mu} S^{i}} < \pi},{\forall{i \in \left\{ {0,\ldots\mspace{14mu},T} \right\}}}} \\0 & {{otherwise}.}\end{matrix} \right.} & (50)\end{matrix}$

Here the contract is knock-in because it only has non-zero payoff if theasset goes below some value π. in the amount the asset is below thestrike price K.

In the examples discussed so far, there has only been one payment datewhere an exchange takes place between the contract issuer and holder, attime T. It is possible for some path-dependent options to have severalpayment dates, where several payments are made at different timesthroughout the course of the contract duration.

Now to introduce the notion of a discounted payoff. As expected, theprice today for any derivative is related to its expected payoff in thefuture. However, the time delay for the payoff to account for theopportunity cost of investing in a risk-free asset with interest rate rcan also be considered. If a contract has a payoff f^(i) at time t_(i)from today, the discounted payoff can be defined as

e ^(−rt) ^(i) f ^(i).  (51)

The price of a derivatives contract is given by the expected value ofthe discounted payoff under the stochastic process for the underlyingassets. In practice, path-dependent derivatives are much more difficultto price computationally and are often priced using Monte Carlosimulations of the paths. This is in contrast to some models forpath-independant derivatives that can even have analytic solutions, suchas the Black-Scholes model for European call options. Path-dependentoptions present an opportunity to use quantum speedups for Monte Carloto gain advantage.

6.4 Auto-Callable Options

A typical example of an auto-callable (‘automatically callable’) optionis a set of binary options, each of which pays different fixed payoutsat different payment dates and then knocks out the whole product (e.g.,voids all future payoffs) if it makes a payout at any of the paymentdates. More formally, let (K, t_(i), f^(i)) be a binary option that haspayoff f^(i) if the underlying asset value is above strike price K attime t_(i). An auto-callable is then a set

{(K,t ₁ ,f ₁),(K,t ₂ ,f ₂), . . . ,(K,T,f _(T))},  (52)

where {t_(i)} and {f_(i)} typically increase linearly. If any of thebinary options (K, t_(i), f_(i)) pays out (is in the money), then allsubsequent options {(K, t_(j), f_(j))}_(j>i) are knocked out (e.g.,voided).

In practice, these binary options are often bundled with a shortknock-in put option, that is, a knock-in put option given to the issuerby the holder, which mitigates risk for the issuer and decreases theprice for the client. As with the set of binary options, this put optionis also knocked out if any of the binary options (K, t_(i), f^(i)) paysout.

As is common with many options, auto-callables can be expanded to havemultiple underlying assets. In this case, it is typical to tie theoverall option payoff to the best or worst performing asset, whereperformance is defined in terms of returns and the payoff of theknock-in put option is proportional to the return of the best or worstperforming asset if that asset is below the strike price. Note that thestrike for the underlying j will often be the same in return space, thatis, the strike price for each asset K_(j) can be written as kS_(j) ⁰where k is the same constant for all underlying assets. Thus, if theworst performing asset is in the money (e.g., above the strike price),then so are all the other assets. Conversely, if the best performingasset is below the strike price (or out of the money), then so are allthe other assets. In principle the different underlying assets couldhave independent strike prices but since this in not common, theassumption can be made that all the strike prices are defined as kS_(j)⁰.

The contingent payoffs and the knock-in put mean that auto-callableshave a payoff that is strongly path dependent. This means that they arecomputationally expensive to price in practice, sometimes taking five toten seconds using classical Monte Carlo methods with at least fortythousand paths.

6.5 Target Accrual Redemption Forwards

A target accrual redemption note (TARN) is any derivative whose payoffis capped at a specified target amount. The term historically referredonly to notes, hence the name, but has now come to include anyderivative with an accrual cap. The various embodiments of the subjectdisclosure uses a commonly used TARN called a target accrual redemptionforward (TARF) as an example case study to implement quantum resourceestimation system 102. A TARF is a set of forwards with a couple ofknock-out conditions. Specifically, it is a derivative with a singleunderlyer with several (e.g., 20-60) payment dates and a forward priceF. Throughout the contract, there can be two fixed strike pricesK_(upper)=F and K_(lower)<F. At each payment date t_(i), there areseveral payoff possibilities:

$\begin{matrix}{f^{i} = \left\{ \begin{matrix}{S^{t_{i}} - F} & {{{if}\mspace{14mu} S^{t_{i}}} > K_{upper}} \\0 & {{{if}\mspace{14mu} K_{{lowe}r}} \leq S^{t_{i}} \leq K_{upper}} \\{a\left( {S^{t_{i}} - F} \right)} & {{{if}\mspace{14mu} S^{t_{i}}} < K_{{lowe}r}}\end{matrix} \right.} & (53)\end{matrix}$

where S^(t) ^(i) is the price of the underlyer at the payment date t_(i)and α is a positive constant. Note that when S^(t) ^(i) <K_(lower), thepayoff is negative and hence the holder of the derivative takes a loss.The constant α makes this loss asymmetric if it happens and is often oneor two.

In addition, a TARF will have two knock-out conditions based on aknock-out threshold π and accrual cap C. The first condition states thatif at any payment date the price of the underlying is greater or equalto π, the derivative contract is immediately knocked out (e.g., withoutpayment for that date). The second condition is if at any payment datet_(i) the total gains of the holder are going to exceed the accrual capC due to the payoff f^(i), the contract holder instead only receives theamount such that their total gains sum up to C and the contract is thenknocked out.

7.0 Insufficiency of Grover-Rudolph Loading

The Grover-Rudolph algorithm is often cited as a method to efficientlycreate quantum superpositions that correspond to classicaldistributions. For a given probability distribution {p_(i)} of a randomvariable x, the algorithm creates a quantum superposition of the form

|ψ({p _(i)})

=Σ_(i)√{square root over (p _(i))}|i

.  (54)

The algorithm is inductive in nature and starts by assuming that thereis a way to divide the probability distributions into some number 2^(m)of regions in the domain of interest and create the state

|ψ_(m)

=Σ_(i=0) ² ^(m) ⁻¹ √{square root over (p _(i) ^((m)))}|i

,  (55)

where p_(i) ^((m)) is the probability for the random variable to lie inregion i. Then it aims to add one qubit to the state of Equation (55),to further subdivide the 2^(m) regions into a 2^(m+1) discretization ofthe probability distribution with an evolution of the form

√{square root over (p _(i) ^((m)))}|i

→√{square root over (α_(i))}|i

|0

+√{square root over (β_(i))}|i

|1

,  (56)

where α_(i)(β_(i)) is the probability for the random variable to lie inthe left or right half of region i. Letting x_(L) ^(i) and x_(R) ^(i)denote the left and right boundaries of region i, the function

$\begin{matrix}{{f(i)} = \frac{\int_{x_{L}^{i}}^{\frac{x_{R}^{i} - x_{L}^{i}}{2}}{{p(x)}{dx}}}{\int_{x_{L}^{i^{R}}}^{X}{{p(x)}{dx}}}} & (57)\end{matrix}$

is the probability that, given x lies in region i, it also lies in theleft half of the region. If a circuit can be constructed that canperform the computation

√{square root over (p _(i) ^((m)))}|i

|0 . . . 0

→√{square root over (p _(i) ^((m)))}|i

|θ _(i)

  (58)

with θ_(i)=arccos√{square root over (f(i))}, then a controlled rotationof angle θ_(i) on the m+1th qubit yields

√{square root over (p _(i) ^((m)))}|i

|0

→√{square root over (p _(i) ^((m)))}|i

|θ _(i)

(cos θ_(i)|0

+sin θ_(i)|1

).  (59)

After uncomputing |θ_(i)

,

|ω_(m+1)

=Σ_(i=0) ² ^(m+1) ⁻¹√{square root over (p _(i) ^((m+1)))}|i

,  (60)

which is the extension of the state in Equation (55) to one extra qubit.Performing this iteration n=log₂N times gives a discretization of thedistribution over N total number of points across n qubits.

In practice, the efficiency of the Grover-Rudolph method relies on theability to perform the integrals in Equation (57) in superposition. Theargument in the original formulation is that probability distributionsthat can be integrated efficiently classically using probabilisticmethods (e.g., using Monte Carlo) can be equivalently efficientlyintegrated quantumly. However, since the ultimate goal in quantumderivative pricing is to provide a faster alternative to Monte Carlointegration over a probability distribution, performing this integral aspart of an initial state preparation without any corresponding quantumspeedup, nullifies the advantage offered by amplitude estimation as analternative to Monte Carlo. While efficient from a complexity point ofview, this means that Grover-Rudolph is insufficient as a method forquantum advantage in derivative pricing.

More recently, an approximate method to implement the Grover-Rudolphalgorithm for standard normal probability distributions was presented,where the authors suggest the expression in Equation (57), written as

$\begin{matrix}{{{g\left( {x,\delta} \right)} = \frac{\int_{x}^{x + {\delta/2}}{{p(x)}{dx}}}{\int_{x}^{x + \delta}{{p(x)}{dx}}}},} & (61)\end{matrix}$

can be approximated as

g(x,δ)≈½+⅛δx+

(δ²),  (62)

for small δ. As the δ parameter decreases with each iteration of theGrover-Rudolph algorithm adding a qubit to the discretization, theauthors highlight that for m≥7 the approximation in Equation (62)becomes sufficiently accurate. However, because the Grover-Rudolphconstruction is iterative, the m<7 terms can be computed before theabove approximation becomes possible. As such, the integrals in Equation(57) are computed classically and then loaded into the correspondingquantum registers. While this approximation allows the simplification ofthe general Grover-Rudolph algorithm for standard normal distributionsafter a certain point in the iteration, it does not change the fact thatit can involve computing integrals over the entire domain of theprobability distribution, thus making it practically infeasible for thesame reason as the original Grover-Rudolph method.

8.0 Fixed-Point Quantum Arithmetic Resources

This section describes preliminaries for common quantum arithmeticoperations and the synthesis of arbitrary rotations. These operationsare used in resource estimation and error analysis. Quantum arithmeticcan be involved in path loading using the Riemann summation method(Section 3.1) and the re-pararameterization method (Section 3.2), aswell as the payoff calculation described in Section 4.0. For the Riemannsum method, all the arithmetic operations involved in Equation (12) canbe performed, as well as computation of the arcsine and square root of aquantum register for the payoff calculation in Equation (15). Algorithmsto perform individual arithmetic operations efficiently have beenidentified, where resources are usually reported as a number of Toffoligates or T-gates. In cases where arithmetic algorithms are performedfrom previous work in the literature, the gate cost in terms of the gateset reported by the authors has been reported herein.

In the fault-tolerant setting, estimation of the T-depth of the circuitsin a Clifford+T gate set decomposition can be made and the assumptioncan be made that Toffoli gates can be constructed with a T-depth of oneusing ancilla qubits. For each operation the assumption can be made thatthe resulting circuits can be parallelized wherever possible.

8.1 Resource Estimation

Perform all calculations in fixed-point arithmetic. An n-bitrepresentation of a number x is

$\begin{matrix}{{x = {\underset{\underset{p}{︸}}{x_{n - 1}\mspace{14mu}\ldots\mspace{14mu} x_{n - p}}.\underset{\underset{n - p}{︸}}{x_{n - p - 1}\mspace{14mu}\ldots\mspace{14mu} x_{0}}}},} & (63)\end{matrix}$

where x_(i)ϵ0,1 denotes the i-th bit of the binary representation of xand p denotes the number of bits to the left of the binary decimalpoint. The choice of n and p controls the error that can be allowed ineach calculation as well as the resources that can be used to performarithmetic on the registers. Once the values of (n, p) are chosen sothat the overall arithmetic error is acceptable for the problem underconsideration, keep them constant throughout the analysis. It ispossible to tailor these values for different components of the circuitand reduce the overall resources that can be used, but for simplicity ofdescription in the subject disclosure, this potential optimization canbe ignored.

Let TF_(f) and T_(f) denote the number of Toffoli gates and the T-depthused to compute an arithmetic function or logical operation f. Theestimates for the operations are functions of the fixed-point registersize (n, p) that will be used to represent the underlying quantum statesinvolved in the computations.

Addition and/or Subtraction

Perform addition of two n-qubit registers in place with Toffoli cost 2n−1. Note that subtraction is given by a−b=˜(˜a+b) and so can beimplemented as an addition with 2n extra X gates, which does not changethe Toffoli count.

Consider the T-depth cost of controlled and uncontrolled addition, whereaddition circuits can be constructed with T-depth T_(add)=10 independentof the register size, and controlled addition with T-depth cost of

(n) for registers of size n.

Multiplication

For multiplication, use a controlled addition circuit and a Toffolicount of

TF _(mul)(n,p)= 3/2z ²+3np+ 3/2n−3p ²+3p.  (64)

This method can also be used for division of a quantum register by aclassical value, which can be done by inverting the classical value andemploying the multiplication algorithm.

The fixed-point multiplication method involves n controlled additions,which thus has T-depth cost of

(n²). These methods can use ancilla qubits proportional to the registersize, but the circuits include uncomputing the ancillas, meaning thatthey can be reused for each subsequent addition that is not done inparallel. Because the computations can be parallelized across the dassets and T timesteps, include an additional T*d*n qubits when countingthe total to account for these potential ancilla qubits.

Additionally parallelize each multiplication circuit, by considering theregister of one factor as z≥1 independent registers of size n/z, andeach controlled addition can happen in parallel for the z subregisters.This can use n·(z −1) extra qubits and z −1 additions to accumulate thez sub-results into the final result. z=1 denotes that no extraparallelization is employed. If the pairwise accumulation additions canbe parallelized as well, a total T-depth cost of parallelizedfixed-point multiplication can be given by

$\begin{matrix}{{T_{mul}\left( {n,z} \right)} = {{\left\lceil \frac{n}{z} \right\rceil \cdot n} + {\left\lceil {\log_{2}z} \right\rceil \cdot {T_{add}.}}}} & (65)\end{matrix}$

Square Root

Employ a square root algorithm that can be extended for quantumregisters in fixed-point representation. For an (n, p)-sized number x,compute √{square root over (x)} by treating x as an n-digit integer, andthen shifting the result to the right (n−p)/2 times. This amounts toperforming

$\begin{matrix}{\left. \sqrt{x}\mapsto\sqrt{\frac{x*2^{n - p}}{2^{n - p}}} \right..} & (66)\end{matrix}$

The Toffoli count of this square root algorithm is

$\begin{matrix}{{{TF}_{sq}\left( {n,p} \right)} = {\frac{n^{2}}{2} + {3n} - 4.}} & (67)\end{matrix}$

The T-depth of this algorithm as reported by the authors is givenT_(sq)(n)=5n+3 and can use 2n+1 qubits.

Logical Operations

For comparisons between quantum registers or between a quantum registerand a constant, use a logarithmic comparator with Toffoli/T-depth of2└log₂(n−1)┘+5, which includes uncomputing the intermediate ancillas.The logical OR operation for a 2-qubit input can be performed with aToffoli/T-depth of one.

Exponential

A generic quantum algorithm can be used to calculate smooth classicalfunctions using a parallel piecewise polynomial approximation. Applythis to estimate the resources of computing exponentials. The algorithmtakes parameters k and M, which control the polynomial degree chosen forthe piecewise approximations and the number of domain subintervalsrespectively. The total number of Toffolis is given by

TF _(exp)(n,p,k,M)= 3/2n ² k+3npk+ 7/2zk−3p ² d+3pk−d+2Md(4┌log₂M┐−8)+4Mn.  (68)

This algorithm, which can also be used to compute the arcsine function,involves k iterations of a multiplication and an addition, wherek-degree polynomials are used for the approximation. Additionally, for Mchosen subintervals, it involves M comparison circuits between then-qubit input register and a classical value. Using a comparator withT-depth of 2└log₂(n−1)┘+5, the T-depth of a parallel polynomialevaluation circuit is

T _(pp)(n,z)=k(T _(mul)(n,z)+T _(add))+M(2└log₂(n−1)┘+5),  (69)

where z is the optional parallelization factor introduced in theresource estimation above for the multiplication circuit.

The qubit count for the parallel polynomial evaluation scheme forchoices of the polynomial degree k and number of subintervals M is

q _(pp)(n,k,M)=n(d+1)+┌log₂ M┐+1.  (70)

Arcsine

To calculate the arcsine, employ the above described generic quantumalgorithm just as can be done for the exponential. However, because thederivative

$\begin{matrix}{\frac{d\;{\arcsin(x)}}{x} = \frac{1}{\sqrt{1 - x^{2}}}} & (71)\end{matrix}$

diverges near ±1, the authors of the generic quantum algorithm use thetransformation

$\begin{matrix}{{\arcsin(x)} = {\frac{\pi}{2} - {2{\arcsin\left( \sqrt{\frac{1 - x}{2}} \right)}}}} & (72)\end{matrix}$

to handle the interval xϵ[0.5,1]. Since the computation of the arcsinecan involve a conditional square root evaluation of the argument and,whenever an arcsine is calculated, calculate the square root as well e.Equation (15)), and use the transformation

$\begin{matrix}{{{\arcsin\left( \sqrt{x} \right)} = {\frac{\pi}{2} - {\arcsin\left( \sqrt{1 - x} \right)}}}.} & (73)\end{matrix}$

The resource estimation considerations then follow similarly to those ina prior art approach. Obtain:

-   -   A comparator to check if x<0.25 (√{square root over (x)}<0.5)        that indicates whether the above transformation should be        applied, which can use two Toffoli gates assuming the value in        the quantum register is normalized.    -   A conditional subtraction and conditional copy depending on the        comparator value above to either prepare √{square root over (x)}        or √{square root over (1−x)}. A conditional copy can use n        Toffolis, a conditional subtraction can use TF_(add)+n Toffoli        gates.    -   TF_(sq) Toffoli gates for the square root computation.    -   The Toffoli gates used for the polynomial evaluation to compute        the arcsine.    -   A conditional copy and conditional subtraction depending again        on the comparator result from the first step, to get either        arcsin(√{square root over (x)}) for x<0.25 or        π/2−arcsin(√{square root over (1−x)}) otherwise.

With the above considerations and the Toffoli count for the polynomialapproximation of arcsin(x), the total Toffoli count for computing|arcsin√{square root over (x)}

is

$\begin{matrix}{{T{F_{arcsq}\left( {n,p,k,M} \right)}} = {{k\left( {{\frac{3}{2}n^{2}} + {n\left( {{3p} + \frac{7}{2}} \right)} - {3\left( {p - 1} \right)p} - 1} \right)} + \frac{n^{2}}{2} + {11n} + {2M{d\left( {{4\left\lceil {\log_{2}M} \right\rceil} - 8} \right)}} + {4Mn} - 2.}} & (74)\end{matrix}$

The T-depth for computing arcsin(√{square root over (x)}) of a number xrepresented in a register of size (n, p), calculated similarly to theexponential is

T _(arcsq)(n,p,z)=T _(sq)(n)+T _(pp)(n,z)+8n+6,  (75)

where T_(sq)(n)=5n+3 is the T-depth for the square root algorithm.

The operation will involve q_(arcsq) qubits, where the qubit criteriafor the arcsine can be given by Equation (70) for a choice of k and M,and 2n+1 for the square root operation

q _(arcsq)(n,k,M)=q _(pp)(n,k,M)+2n+1  (76)

R_(y)

Use R_(y)(θ) rotations in the variational preparation of Gaussiansdiscussed in Section 3.2.1 and controlled-R_(y) rotations to encode thepayoff into the amplitude of an ancilla in Equation (16) as well as thetransition probabilities in the Riemann summation method. Using theRepeat-Until-Success method, an arbitrary single-qubit unitary can beperformed within precision ϵ with a T-depth of approximately 1.15log₂(1/ϵ) using one ancilla qubit and measurement.

When the angle θ to rotate is stored in a separate register |θ

, obtain and/or use a series of R_(y)(θ_(k)) rotations, each controlledon the kth qubit of |θ

where

$\begin{matrix}{{\theta_{k} = \frac{2^{k}}{2^{n - p}}}.} & (77)\end{matrix}$

A single controlled-R_(n) can be performed with an R_(n)-depth of one,R_(n)-count of 3 and with a single ancilla qubit using a decomposition.However each rotation contributes an error ϵ so if |θ

is an n-qubit register (e.g., with p bits to the left of the binarypoint), the end-to-end operation can be performed to precision ϵ withT-depth of at most 1.15n log₂(n/ξ). Reduce this depth slightly bynoticing that the amplitude increase due to any controlled-R_(n)rotation where θ_(k)<arcsin(ϵ) is less than ϵ and hence is uneccesary.Therefore using that observation and Equation (77), compute the totalnumber of rotations to be n−max(└log₂(arcsin(ϵ)┘+(n−p),0). This gives afinal T-depth for a controlled-Ry(θ) operation of

T _(Ry)(n,p,ξ)=1.15ñ log₂(ñ/ξ)  (78)

where ñ=n−max(└log₂(arcsin(E)┘+(n−p),0).

8.2 Error Analysis

Given the fixed-point representation of Equation (63), each arithmeticoperation involving registers results in some approximation error,depending on the specific method used. Outlined here is the arithmeticerror associated with each of the operations described in the previoussection.

Addition and/or Multiplication

Use a fixed-point addition and a multiplication method, where theaddition of two (n, p)-sized registers introduces an error bounded by

${\epsilon_{A} = \frac{1}{2^{n - p}}},$

and the error associated with multiplication is at most

$\begin{matrix}{{\epsilon_{M}\left( {n,p} \right)} = {\frac{n}{2^{n - p}}.}} & (79)\end{matrix}$

For (n,p)-sized registers X and Y, where each register already containsadditive errors ϵ_(X), ϵ_(Y) and each factor X, Y is bounded above by b,the error in the computation of X·Y is given by

ϵ_(mul) =b*(ϵ_(X)+ϵ_(Y))+ϵ_(X)ϵ_(Y)+ϵ_(M)(n,p)  (80)

Exponential

Employ parallel polynomial evaluation methods to estimate the resourcesand associated error in computing exponentials. The error associatedwith the algorithm depends on choices for the degree of the polynomialapproximation and the number of subintervals chosen, but the authors ofsuch algorithm provide explicit error estimates and correspondingresources for errors ranging from 10⁻⁵ to 10⁻⁹. Use these in the overallerror estimate. Compute the exponential of a register that itselfcontains arithmetic error ξ. Denoting the error in computing theexponential of a register ϵ_(exp), the total arithmetic error incomputing the exponential of a register can be approximated to firstorder in ξ in the Taylor expansion of exp(−x+ξ) as

ϵ _(exp)≲ϵ_(exp)+ξ.  (81)

Square Root

As discussed in the previous section, for square root computationsconsider a square root algorithm, extended for quantum registers infixed-point representation. The mapping in Equation (66) can introduce amaximum error of

$\begin{matrix}{\epsilon_{sq} = {\frac{1}{2^{{({n - p})}/2}}.}} & (82)\end{matrix}$

When computing the square root of a register x which already contains(e.g., positive) additive error ξ, the total additive error from thesquare root operation is bounded by ϵ_(sq)+√{square root over (ξ)}. Thisis easily seen by observing that with a square root algorithm whichgives an estimate {circumflex over (x)} with |√{square root over(x)}−{circumflex over (x)}|≤ϵ_(sq), then

|√{square root over (x+ξ)}−{circumflex over (x)}|≤|√{square root over(x)}−{circumflex over (x)}|+√{square root over (ξ)}

≤ϵ_(sq)+√{square root over (ξ)}

where the first inequality follows from (√{square root over(x)}+√{square root over (ξ)})²=x+ξ+2√{square root over (xξ)}≥x+ξ, forpositive x and ξ.

Arcsine

For the arcsine calculation again use the polynomial evaluation method,where the sample resource estimates for error rates range from 10^(−s)to 10⁻⁹. Bound the error from the computation of arcsine on a registercontaining an arithmetic error f to begin with. As discussed in Section8.1 compute arcsin(x) for x≤0.5. In addition, when computing thefunction arcsin(x) in the algorithms of the subject disclosure describedherein, only do it for x≥0. This gives a domain of 0≤x≤0.5 for arcsin(x)error calculation. Given this domain, notice that the slope of arcsin(x)is always monotonically increasing with a maximum at x=0.5. Thereforecomputing the error when x=0.5 gives the upper bound:

ϵ _(arcsin)≤|arcsin(0.5)−arcsin(0.5−ξ)|+ϵ_(arcsin),  (83)

where ϵ_(arcsin) is the error from the computation of the arcsine, givena choice of polynomial degree and number of subintervals.

Sine

As discussed in the previous section, compute the sin(θ) function with aseries of controlled-Ry rotations controlled on qubits from a registercontaining the angle θ. Bound the error from the computation of sin(θ)when the register that is supposed to represent θ is actuallyrepresenting θ+ξ due to an arithmetic error. To quantify the upperbound, notice that in the domain of 0≤θ<π/2, the slope of sin(θ) ismonotonically decreasing, and therefore has a maximum slope at θ=0.Therefore computing the error when θ=0 gives the upper bound:

ϵ_(sin)≤|sin(0+ξ)−sin(0)|+ϵ_(sin)

≤ξ+ϵ_(sin)  (84)

where the inequality sin(a+b)≤sin(a)+b for b≥0 has been used and whereϵ_(sin) is the error arising from the gate decomposition of the Ryoperator described in Section 8.1.

9.0 Riemann Summation

9.1 Riemann Summation Path Loading Resource Estimates

This section examines the T-depth and qubit count to compute Equation(12) in a quantum register, and encode that value into the amplitude ofan ancilla qubit as described in Algorithm 3.1. The calculation is donein log-return space (with reference to Section 1.2) and it involves theresource estimates for the operations described in Section 8.1.

Let T_(f) and q_(f) denote the T-depth and qubit count for an operationf respectively. Assuming the computation across the d assets and Ttimesteps can be parallelized wherever possible, the contributions tothe resources for computing |arcsin√{square root over (P({right arrowover (R)}))}

with P({right arrow over (R)}) given by Equation (12) are

-   -   T_(add) for computing the terms (R−μ) which can be done in        parallel for d assets and T timesteps, where T*d*n qubits are        used to hold the log-returns R for all assets and timesteps.    -   T_(mul) for all R² terms in the expansion of Equation (33)        (e.g., in parallel for all d and T), obtaining and/or using        T*d*n additional qubits.    -   T_(mul)*d₂/(d/2) for all R_(i)R_(j) terms in the expansion of        Equation (33) (e.g., parallel in d, T) and T*d₂*n qubits.    -   T_(add)*┌log(d₂+d)┐ to sum all the terms in Equation (33) in        parallel. The qubits from the previous step can be reused here.    -   T_(exp) to calculate the exponential in Equation (8), using        q_(exp) extra qubits with g_(exp) given by Equation (70) for a        choice of parameter values determined by the desired        approximation accuracy.    -   T_(arcsq) to calculate the arcsin and square root in        |arcsin√{square root over (P({right arrow over (R)}))}        , with qubit resources given by Equation (76).    -   T_(add)*(T−1) and (T−1)*d*n qubits to calculate all the sums        R_(j) ^(t=1)+R_(j) ^(t=2)+ . . . +R_(j) ^(t=t′) for t′ϵ[2,T] in        Equation (11).    -   T_(ecp) to calculate the prices across all assets and all        timesteps in Equation (11) in parallel, using g_(exp)*d*T more        qubits.    -   A T-depth of 1.15n log₂(n/ϵ) to perform the ancilla rotation to        precision E, controlled on the register where |arcsin√{square        root over (P({right arrow over (R)}))}        is computed. This involves n ancilla qubits using the        controlled-R_(y) decomposition.

Moreover, include an additional register of size T*d*n to implement theaddition circuit with constant T-depth and (z −1)*T*d extra qubits ifthe parallel multiplication scheme described in Section 8.1 is used tocalculate prices across assets and timesteps, where z≥1 is the optionalparallelization factor chosen. Note that extra qubit counts to computethe (R −μ) terms and the sum in Equation (33) have not been includedbecause these can be done in place using the existing registers to holdeach R_(i). This is possible because after computing the sums andexponentials in Equation (11) (e.g., which can be done before computingthe sums) the values of R_(i) are not used again.

The total T-depth of the Riemann summation path loading process toprecision ϵ for d assets and T timesteps using registers of size (n, p)is then

T _(RS)(n,p,d,T,ϵ)=n ²+2n ² d ₂ /d+10(d ₂ +d)+10T+9n+5+1.15nlog₂(n/ϵ)+2T _(exp)(n,p,ϵ)+T _(arcsin)(n,p,ϵ),  (85)

where the dependency of T_(exp) and T_(arcsin) on ϵ denotes that thepolynomial approximation parameters k and M in Equation (69) for eachfunction will depend on the target accuracy of the process. The totalnumber of qubits involved is

q _(RS)(n,p,d,T,ϵ)=Tn(4d+d ₂)+3n+1+q _(exp)(n,p,ϵ)(1+dT)+q_(arcsin)(n,p,ϵ).   (86)

9.2 Importance Sampling for Normalization in Riemann Summation

This section introduces a technique closely related to classicalimportance sampling to overcome the problem of the exponentiallyincreasing scaling shown in Algorithm 3.1. The main idea is toapproximate the target distribution by another distribution that can beloaded efficiently and then use quantum arithmetic only to adjust forthe (e.g., multiplicative) error.

Suppose a univariate probability density function f:[0,1]→[0, P], withP>1 and ∫_(x=0) ¹f(x)dx=1 and a payoff function g:[0,1]→[0,1]. In theconsidered context, g will be applied only once. Thus, it can be assumedthat it takes values in [0,1] without changing the overall complexity ofthe approach. As introduced before, consider the scaled function f(x)/Pand a corresponding operator

, as well as a corresponding operator

to prepare a state on n+2 qubits given by

$\begin{matrix}{\left. \left. {{{\left. \left. {{{\left. {{\frac{1}{\sqrt{N}}\sum\limits_{i = 0}^{N - 1}}\; ❘i} \right\rangle_{n}\left( \sqrt{1 - {{f\left( x_{i} \right)}/P}} \middle| 0 \right\rangle} + \sqrt{{f\left( x_{i} \right)}/P}}❘1} \right\rangle \right)\left( \sqrt{1 - {g\left( x_{i} \right)}} \middle| 0 \right\rangle} + \sqrt{g\left( x_{i} \right)}}❘1} \right\rangle \right),} & (87)\end{matrix}$

where x_(i)=i/N. Then, the probability of measuring |11

in the last two qubits is given by

$\begin{matrix}{{\frac{1}{PN}{\sum\limits_{i = 0}^{N - 1}\;{{f\left( x_{i} \right)}{g\left( x_{i} \right)}}}},} & (88)\end{matrix}$

and when multiplied with P corresponds to the Riemann sum approximating

∫_(x=0) ¹ f(x)g(x)dx=

[g(X)] for X˜f.

Further, consider a probability distribution h(x_(i))ϵ[0,1] that can beefficiently loaded into a quantum state, that is, where it is known howto efficiently construct a quantum operator H such that

|0

_(n)=Σ_(i=0) ^(N−1)√{square root over (h(x _(i)))}|i

_(n).  (89)

Suppose there exists h such that f(x)/(h(x)N)ϵ[0,1] for all x, thenconstruct a new operator

_(h) defined as

_(h) :|i

_(n)|0

|i

_(n)(√{square root over (1−f(x _(i))/(h(x _(i))N))}|0

+√{square root over (f(x _(i))/(h(x _(i))N))}|1

).  (90)

Combining

and

_(h) leads to

_(h)

|0

_(n)|0

=Σ_(i=0) ^(N−1)√{square root over (h(x _(i)))}|i

_(n)( . . . +√{square root over (f(x _(i))/h(x _(i))N))}|1

)( . . . +g(x _(i))|1

),  (91)

which implies a probability of measuring |11

in the last two qubits given by

$\begin{matrix}{{\frac{1}{N}{\sum\limits_{i = 0}^{N - 1}{{f\left( x_{i} \right)}{g\left( x_{i} \right)}}}},} & (92)\end{matrix}$

that is, the Riemann sum approximating ∫_(x=0) ¹f(x)g(x)dx=

[g(X)] for X˜f. Thus, if such a probability distribution h can be found,construct a state that directly corresponds to

[g(X)] without rescaling by multiplying P. It can be easily seen thatfor P≤1, h(x)=1/N can be set to recover the original approach withoutimportance sampling.

In case of multivariate probability density functions, three cases aredistinguished. First, separable functions that can be written as aproduct of univariate functions f_(t) for t=0, . . . , T. In this case,the univariate approach can be applied directly and a correspondingh_(t) for each f_(t) can be found. Second, non-separable multivariateprobability density functions f:[0,1]^(d)→[0,P], with P>1 and∫_(xϵ[0,1]) _(d) f(x)dx=1. Suppose each dimension is discretized using nqubits, that is, have in total N^(d) grid points. Then, find aprobability distribution h such that f(x)/(h(x)N^(d))ϵ[0,1] for all x,and the analysis is analog to the univariate case. Last, consider thecase of a multivariate probability density function coming from astochastic process and given by

f(x ₀ , . . . x _(T))=f ₀(x ₀)Π_(t=1) ^(T) f _(t)(x _(t) |x_(t-1)),  (93)

where x_(t) ϵ[0,1]^(d) and f₀(x₀),f_(t)(x_(t)|t_(x-1))ϵ[0, P] for t=0, .. . , T. Suppose a separable probability distribution

h(x ₀ , . . . ,x _(T))=Π_(t=0) ^(T) h _(t)(x _(t)),  (94)

that can be loaded efficiently as well as a corresponding decompositionh_(t)(x_(t))=h_(t) ^(t)(x_(t))h_(t) ^(t+1)(x_(t)), with h_(T)^(T+1)(x)=1. Then,

$\begin{matrix}{\frac{f\left( {x_{0},\ldots\mspace{14mu},x_{T}} \right)}{N^{{({T + 1})}d}} = {\frac{f_{0}\left( x_{0} \right)}{{h_{0}^{0}\left( x_{0} \right)}N^{d}}{h_{0}^{0}\left( x_{0} \right)}{h_{0}^{1}\left( x_{0} \right)}{\prod\limits_{t = 1}^{T}\;{\frac{f_{t}\left( {x_{t}❘x_{t - 1}} \right)}{{h_{t - 1}^{t}\left( x_{t - 1} \right)}{h_{t}^{t}\left( x_{t} \right)}N^{d}}{h_{t}^{t}\left( x_{t} \right)}{{h_{t}^{t + 1}\left( x_{t} \right)}.}}}}} & (95)\end{matrix}$

Thus if h is found such that the individual h_(t) can be efficientlyloaded and

$\begin{matrix}{{\frac{f_{0}\left( x_{0} \right)}{{h_{0}^{0}\left( x_{0} \right)}N^{d}} \in \left\lbrack {0,1} \right\rbrack},{\forall x_{0}}} & (96) \\{{\frac{f_{t}\left( {x_{t}❘x_{t - 1}} \right)}{{h_{t - 1}^{t}\left( x_{t - 1} \right)}{h_{t}^{t}\left( x_{t} \right)}N^{d}} \in \left\lbrack {0,1} \right\rbrack},{\forall x_{t - 1}},x_{t},{t = 1},\ldots\mspace{14mu},T,} & (97)\end{matrix}$

then, efficiently load the stochastic processes without the exponentialscaling overhead P^(T+1). Again, it can be easily seen that for P≤1,h_(t)(x_(t))=h_(t) ^(t)(x_(t))=1/N^(d) can be set and h_(t)^(t+1)(x_(t))=1 to recover the original approach without importancesampling.

Note that even though an h that satisfies all criteria may not always befound, this approach can still help to lower the overhead coming fromscaling.

10.0 Re-Parameterization Path Loading Resource Estimates

To prepare the standard normal distributions that can be used by quantumresource estimation system 102 in implementing the re-parameterizationloading approach of the subject disclosure, quantum resource estimationsystem 102 (e.g., via variational component 202) can employ thevariational method described in Section 3.2.1 and the corresponding gateand/or qubit cost depending on the desired accuracy of theapproximation. In addition to that, the cost of computing the affinetransformation {right arrow over (R)}^(t)={right arrow over(μ)}^(t)+L^(T) {right arrow over (R)} ^(t) as described in Algorithm 3.2will also be incurred. Note that the affine transformation is used tocalculate the asset prices from the log-returns, which for asset j attime t′ will be

S _(j) ^(t′) =S _(j) ^(t=0) e ^(μ) ^(j) ^(t′+Σ) ^(i=0) ^(d) ^(L) ^(ji)^(T) ^(R) ^(i) ^((t′)) =e ^(ln S) ^(j) ^(t=0) ^(+μ) ^(j) ^(t′+) ^(Σ)^(i=0) ^(d) ^(L) ^(ji) ^(T) ^(R) ^(i) ^((t′)),  (98)

where R _(i)(t′) is the ith component of the sum ({right arrow over(R)}(t=1)+{right arrow over (R)}(t=2)+ . . . +{right arrow over(R)}(t=t′)). One complication in Equation (98) is that each asset pricecannot be computed fully in parallel across the d assets, because thelog-returns of any correlated assets will contribute to the computationof each other's price. In the case where all assets are pairwisecorrelated, quantum resource estimation system 102 can compute thecontributions to each asset's price from the log-returns of all d assetsat that timestep, requiring in total d² additions to compute all assetprices per timestep. However, quantum resource estimation system 102 canperform d additions in parallel where the contribution of asset j'sreturn to the price of asset (j+i) % d is computed for a choice of iϵ[0,d −1], since all d such operations have distinct source and targetregisters. Then d rounds of additions will compute the term Σ_(i=0)^(d)L_(ji) ^(T) R _(i)(t′) for all assets, and if quantum resourceestimation system 102 computes ({right arrow over (R)}(t=1)+{right arrowover (R)}(t=2)+ . . . +{right arrow over (R)}(t=t′)) in a separateregister for each t′, the above calculation can be also parallelizedacross all timesteps.

The arithmetic error in computing Equation (98) can be reduced (e.g.,minimized) by increasing the qubit register sizes to accommodate thelargest values possible for the sums over the timesteps T and assets d.If each gaussian prepared in Equation (38) is discretized using nqubits, then n+┌log₂ T┐ qubits will be enough to hold the largest valueof the expression {right arrow over (R)}(t=1)+{right arrow over(R)}(t=2)+ . . . +{right arrow over (R)}(t=t′)). An additional ┌log₂ d┐qubits will achieve the same for Σ_(i=0) ^(d)L_(ji) ^(T) R(t′), assumingthe coefficients |L_(ji) ^(T)|≤1 for all i,j. This condition is not hardto satisfy for typical situations of practical interest, which can beargued by looking at the elements of the covariance matrixΣ_(ij)=Δtρ_(ij)σ_(i)σ_(j) (e.g., where by definition |ρ_(ij)≤1).Typically, annualized volatilities are smaller than 100% (i.e. σ_(i)<1)and the timestep usually satisfies Δt<1, meaning the price of theunderlying assets can be sampled more frequently than just yearly. Ifneither condition is satisfied however, quantum resource estimationsystem 102 can choose a smaller Δt to ensure |Σ_(ij)|<1, at the cost ofincreasing the number of timesteps in the calculation.

The contributions to the T-depth and qubit count for loading the pathsand computing the asset prices in the re-parameterization approach for aderivative defined on d assets T timesteps are

-   -   T_(R) _(y) (n)·(L+1) T-depth for loading the gaussian states in        Equation (38) using the variational method from Section 3.2.1,        where each Gaussian is prepared in parallel and the variational        ansatz has depth L. This step involves T*d*n qubits where n        qubits are used to prepare each Gaussian state.    -   T_(add)*(T−1) for calculating all the sums ({right arrow over        (R)}(t=1)+{right arrow over (R)}(t=2)+ . . . +{right arrow over        (R)}(t=t′)) for t′ϵ[2, T] in Equation (98), involving an extra        T*d*(n+┌log₂T┐) qubits.    -   T_(add)*d to compute all contributions to Σ_(i=0) ^(d)L_(ji)        ^(T) R _(i)(t′) in Equation (98) and T*d*┌log₂ d┐ more qubits.    -   T_(add) to compute the μ_(j)t′+ln S_(j) ^(t=0) contribution in        Equation (98) across assets and timesteps.    -   T_(exp) to compute the exponential in Equation (98) across        assets and timesteps, and q_(exp)*d*T additional qubits with        q_(exp) given by Equation (70).

All in all, the total T-depth for path loading using there-parameterization method to precision ϵ for d assets and T timestepsis

T _(RP)(n,d,T,L,ϵ)=1.15n log₂(n/ϵ)(L+1)+10(d+T)+T _(exp)(n,ϵ),  (99)

with qubit count

q _(RP)(n,d,T)=(n+n+q _(exp)( n ,ϵ))dT,  (100)

where n=n+┌log₂ T┐+┌log₂ d┌.

11.0 Method for Gaussian Loader Training

This section describes an approximate method to initialize the quantumregister using the Variational Quantum Eigensolver (VQE) approach. Thisalgorithm features a parametrized circuit which in turn produces aparametrized state |ψ({θ})

that approximately represents the target state |φ₀

and updates

its parameters {θ} to optimize the expectation value of a suitable costfunction. Here it is shown that the choice of the cost function tooptimize is crucial for the success of the training.

Energy Based Training

As a first method, quantum resource estimation system 102 can adopt aphysics based approach and define an operator H, such that itsexpectation E value assumes its lowest possible value, E₀, whenevaluated on the target state,

E ₀=

ψ₀ |H|ψ ₀

  (101)

In physics application, the operator H is usually called Hamiltonian, Ethe energy, and |ϕ₀

the ground state. It is well known the Gaussian function

$\begin{matrix}{{\phi_{0}(x)} = {\left( \frac{m}{\pi} \right)^{1/4}e^{- {m{({x - x_{0}})}}^{2}}}} & (102)\end{matrix}$

is the ground state of the quantum harmonic oscillator Hamiltonian

H=P ²2m+m(X−x ₀)²2,  (103)

where X is the position operator in real space, and P=−iddx is themomentum operator, m is a parameter that determines the variance of thedesired Gaussian distribution, and x₀ is the center of gaussiandistribution. In this case, to find a state ϕ₀(x) such that ϕ₀ ²(x)=

(x₀,σ), quantum resource estimation system 102 can tune m to m=1/(2σ²).

Notice that it is always possible to find a generating Hamiltonianfunction such that its ground state is the square root of the smoothdistribution function to be loaded (e.g., the target probabilitydistribution).

To translate such considerations into an operational workflow quantumresource estimation system 102 can define a way to compute theexpectation value of Equation (103) using a quantum computer. To thisend it can be observed that the operator X² is diagonal in thecomputational basis, so it can be measured directly from the bit-stringhistogram counts N_(counts)(j) generated by the repeated wavefunctioncollapses. The operator P² is diagonal in the momentum basis. Thisimplies the addition of a centered Quantum Fourier Transform (QFT)circuit after the state preparation block. Quantum resource estimationsystem 102 can use the centered Fourier transform to allow for negativemomenta. As described above, quantum resource estimation system 102 canwork in a discrete position space x_(i)=−w+iΔx, with i=0, . . . 2^(n)−1,and Δx=2w/2^(n). Without loss of generality quantum resource estimationsystem 102 can choose the domain to be centered in zero. The energy,E=E_(X) ₂ +E_(P) ₂ , can be computed in the following way,

$\begin{matrix}{E_{X^{2}} = {\frac{1}{N_{shots}}{\sum\limits_{j = 0}^{\mathcal{N}}\;{\frac{m}{2}{N_{counts}(j)}\left( {{j \times \Delta\; x} - x_{0}} \right)^{2}}}}} & (104) \\{E_{P^{2}} = {\frac{1}{N_{shots}}{\sum\limits_{j = 0}^{\mathcal{N}}{\frac{1}{2m}{N_{counts}(j)}\left( {j \times \Delta\; p} \right)^{2}}}}} & (105)\end{matrix}$

where N_(shots) is the total number of circuit repetitions for thespacial and momentum basis. N_(counts)(j) (with0≤N_(counts)(j)≤N_(shots), Σ_(j) N_(counts)(j)=N_(shots)) is the numberof measurements that collapsed onto the qubit basis state correspondingto the binary representation of integer j. This strategy bypasses theuse of a Pauli representation of Equation (103), which would include anexponentially increasing number of Pauli string to be measured with thequbit register size.

The first step of the program is to verify numerically the possibilityto prepare a state that is systematically converging to Equation (102),using a quantum circuit. Adopting a variational approach will circumventcostly quantum arithmetic operations at the expense of introducingsources of error which are always present in numerical variationalapproaches. The most trivial one concerns the possibility of gettingtrapped in local minima during the (classical) optimization procedure.The second, and more profound one, is linked with the representationalpower of trial states produced by the (e.g., shallow) quantum circuits.

The main choice for the ansatz is the so-called R_(y)−CNOT circuit. Theinitial state, defined on a n qubits register, which quantum resourceestimation system 102 can set to |0

^(⊗n), is evolved under the action of the unitary U({right arrow over(θ)}) to give the trial wave function |ψ({right arrow over (θ)})

.

The circuit is made of a series of L blocks built from single-qubitrotations U_(R)({right arrow over (θ)}^(k)), followed by an entanglerU_(ENT), that spans the length of the qubit register. In theexperimental case studies described above, quantum resource estimationsystem 102 chose the choice of a ladder of CNOTs gate with linearconnectivity, such that qubit q_(i) is target of qubit q_(i−1) and thecontrols qubit q_(i+1), with i=1, . . . , n−2. One additional layer ofU_(R) gates is applied at the end, such that the number of variationalparameters is n×(L+1).

Since the single-qubit rotations are all local operations, U_(R)({rightarrow over (θ)}^(k)) can be written as a tensor product of the rotationsof a single qubit:

$\begin{matrix}{{{U_{R}\left( {\overset{\rightarrow}{\theta}}^{k} \right)} = {\underset{i = 0}{\overset{n - 1}{\otimes}}{R_{Y}\left( \vartheta_{qi}^{k} \right)}}},} & (106)\end{matrix}$

R_(y)(ϑ_(q) _(i) ^(k)) is a rotation on the Y-axis on the Bloch sphereof qubit q_(i), and k=1, . . . , L+1. The full unitary circuit operationis described by

$\begin{matrix}{{{U\left( \overset{\rightarrow}{\theta} \right)} = {{U_{R}\left( {\overset{\rightarrow}{\theta}}^{L + 1} \right)}\overset{\overset{L\text{-}{times}}{︷}}{U_{ENT}{U_{R}\left( {\overset{\rightarrow}{\theta}}^{L} \right)}\ldots\mspace{14mu} U_{ENT}{U_{R}\left( {\overset{\rightarrow}{\theta}}^{1} \right)}}}},} & (107)\end{matrix}$

and the parametrized state as

|ψ({right arrow over (θ)})

=U({right arrow over (θ)})|0

^(⊗n)  (108)

Notice that the unitary U({right arrow over (θ)}) describes the fullcircuit, but not the pre-measurement change of basis that can beinvolved to collapse the wavefunction in momentum space as explainedabove.

For each n and L parameters quantum resource estimation system 102 canrepeat the optimization runs, for instance, eight times in order togather sufficient statistics, as the optimizations can remain stuck insuboptimal minima. Since quantum resource estimation system 102 can useclassical emulation of the quantum circuits the only source of error inthe optimizations is originated from the classical optimizer. In theexperimental runs performed by quantum resource estimation system 102 toimplement the above described case studies, quantum resource estimationsystem 102 first performed a warm up run with the COBYLA optimizer,followed by a longer run using the BFGS optimizer. To enhance theefficiency the optimizations, the starting point for the VQE run atdepth L, uses the optimal parameters found at previous optimisation atthe depth at L −2 or L −1 when available. Notice that the part of thealgorithm that concerns the classical optimization feedback can begreatly improved, for example, using gradient based methods orimaginary-time inspired update schemes.

L_(∞) Training Refinements

As described above, quantum resource estimation system 102 can use thepre-optimized circuits obtained using the energy optimization method asa starting guess for the re-optimization following the L_(∞) as costfunction. In FIGS. 6A, 6B, and 6C it is shown how the direct L_(∞)optimization always fails to provide acceptable results.

FIGS. 6A, 6B, and 6C illustrate example, non-limiting graphs 600 a, 600b, 600 c, respectively, that can facilitate estimation of quantumresources to calculate an expectation value of a stochastic processusing a re-parameterization method in accordance with one or moreembodiments described herein. Repetitive description of like elementsand/or processes employed in respective embodiments is omitted for sakeof brevity.

In each of graphs 600 a, 600 b, and 600 c depicted in FIGS. 6A, 6B, and6C, respectively, optimization runs obtained with the energy-basedmethod are represented by plot 602. In each of graphs 600 a, 600 b, and600 c depicted in FIGS. 6A, 6B, and 6C, respectively, the direct L_(∞)optimization is represented by plot 604. In each of graphs 600 a, 600 b,and 600 c depicted in FIGS. 6A, 6B, and 6C, respectively, the mixedstrategy where the energy based optimization is further refined usingthe L_(∞) optimization is represented by plot 606. To obtain the dataplotted in each of graphs 600 a, 600 b, and 600 c, quantum resourceestimation system 102 can perform eight independent runs at givenparameters n qubits and L.

The complete outcome of the optimizations are illustrated in FIGS. 7Aand 7B.

FIGS. 7A and 7B illustrate example, non-limiting graphs 700 a and 700 b,respectively, that can facilitate estimation of quantum resources tocalculate an expectation value of a stochastic process using are-parameterization method in accordance with one or more embodimentsdescribed herein. Repetitive description of like elements and/orprocesses employed in respective embodiments is omitted for sake ofbrevity.

Graph 700 a illustrates the L_(∞) norm difference between the preparedand the target distribution as a function of the circuit depth L fordifferent qubit register sizes n. Plotted in graph 700 a are the bestamong the eight independent optimization for each parameter. Plots 702in graph 700 a correspond to optimization performed using the energy ofthe quantum harmonic oscillator as a cost function. Plots 704 in graph700 a correspond to the re-fined optimizations using the L_(∞) as costfunction.

Graph 700 b illustrates the difference in energy of the associatequantum harmonic oscillator model as a function of the circuit depth Lfor different qubit register sizes n. Plotted in graph 700 b are thebest among the eight independent optimization for each parameter. Plots702 in graph 700 b correspond to optimization performed using the energyof the quantum harmonic oscillator as a cost function. Plots 704 ingraph 700 b correspond to the re-fined optimizations using the L_(∞) ascost function. As expected the refinement targeting the L_(∞) does notimprove this quantity. The results of the numerical study illustrated ingraphs 700 a and 700 b show that the convergence to the exact groundstate is exponential in the depth, hence in the number of gateoperations.

Failure of the L_(∞) Norm Direct Optimization

Provided here is an empirical explanation concerning the observedfailure of the direct norm optimization technique. To this end quantumresource estimation system 102 can probe the cost function landscape forboth the methods, the energy based and the direct L_(∞) optimization.Quantum resource estimation system 102 can start from an optimizedparameter configuration {right arrow over (θ)}₀ and perform a cut in theparameter space, following the prescription

{right arrow over (θ)}={right arrow over (θ)}₀+λ{right arrow over(η)}  (109)

where {right arrow over (η)} is an vector containing uniformlydistributed random numbers in the range [−1,1] and λϵ[−π, π] is a scalarwhich parametrizes the deformation form the optimal solution.

FIGS. 8A and 8B illustrate example, non-limiting graphs 800 a and 800 b,respectively, that can facilitate estimation of quantum resources tocalculate an expectation value of a stochastic process using are-parameterization method in accordance with one or more embodimentsdescribed herein. Repetitive description of like elements and/orprocesses employed in respective embodiments is omitted for sake ofbrevity.

Graphs 800 a and 800 b depicted in FIGS. 8A and 8B, respectively,illustrate probing of the cost function landscape for three differentcut direction (e.g. three different realizations of the vector r). Theplots illustrated in the top portions of graphs 800 a and 800 brepresent the cost function landscape for the energy E, while the plotsillustrated in the bottom portions of graphs 800 a and 800 b representthe cost function landscape for the L_(∞) norm computed using threedifferent cuts along the parameter space, and for two different setupn=5,7 and depths L=6,10 respectively.

From graphs 800 a and 800 b, it can be observed that the cost functiondefined by the L_(∞) norm is much more corrugated than the one definedby energy E of the associate quantum mechanical toy problem, whichinstead displays a smoother surface. The basins of attraction of theenergy cost-function and the L_(∞) cost-function are overlapping, thishappens because the ground state of the physical problem is close to theGaussian function to be achieved, so the second optimization with theL_(∞) norm does not remain stuck in high-cost local minima outside suchbasin.

Variational Parameters Digitization

While the above described numerical results provide evidence for arather efficient Gaussian state preparation in terms of circuit depthsfor a parametrized circuit, an additional step has to be made in view ofa fault-tolerant implementation of such circuits. In this new-framework,the continuous rotation RY gate can be expanded as a finite product ofdiscrete operations. Following again the Solovay-Kitaev theorem, or morespecialized results, it is possible to have also an efficientrepresentation of any SU(2) operator with a sequence of Clifford+T-gatesthat scale logarithmically with the threshold error E. It can beinvestigated how the results obtained before can be transferred in thisregime where rotation's angles can take only discretized value.Therefore, it can be assumed that each parameter ϑ_(q) _(i) ^(k) canonly be represented by the format i*2π/M_(digit), where i is an integer.

Quantum resource estimation system 102 can adopt a protocol to optimizethe parameters on an a grid. First quantum resource estimation system102 can project the original continuous values parameters on the gridtaking for each parameter the closest grid points. Subsequently quantumresource estimation system 102 can perform a local search on the grid tofind a better combination of the digitized parameters which reduce(e.g., minimize) the L_(∞) norm difference compared to the targetdistribution. Quantum resource estimation system 102 can numericallyshow that the error introduced by such digitization decreasessystematically with the mesh size. If quantum resource estimation system102 considers the L_(∞) norm difference error introduced by suchdigitization, it decreases as 1/Ma_(digit). It can be observed that inall cases, quantum resource estimation system 102 can obtain valuescompatible with the continuous valued solution, or even better, when themesh size reach M_(digit)˜10⁵, this equal in discretizing the space with2π/M_(digit)≈0.0001 radians (rad).

FIGS. 9A, 9B, and 9C illustrate example, non-limiting graphs 900 a, 900b, and 900 c, respectively, that can facilitate estimation of quantumresources to calculate an expectation value of a stochastic processusing a re-parameterization method in accordance with one or moreembodiments described herein. Repetitive description of like elementsand/or processes employed in respective embodiments is omitted for sakeof brevity.

Graph 900 a depicted in FIG. 9A illustrates the L_(∞) norm differencebetween the prepared and the target distribution as a function of thedigitization mesh size M_(digit) for two different circuit depts L andfor the n=4 qubit case. For each M_(digit) quantum resource estimationsystem 102 can “digitize” the eight parameter sets obtained by theprevious independent optimizations (e.g., done considering a continuousdomain for rotation angle values). The empty (e.g., hollow) square andcircle symbols represent to the full dataset, while the solid symbolsrepresent minumum values in the set. Horizontal lines 902 denote thebest values obtained in the previous optimizations done considering thea continuous domain for rotation angle values for each L parameters. Insome examples, the digitization helps in escaping local minima andachieve slightly better solutions. Diagonal lines 904 are aguide-to-the-eye and represent the functions 1/M_(digit) and0.1/M_(digit).

Graph 900 b depicted in FIG. 9B comprises an example, non-limitingalternative embodiment of graph 900 a, where graph 900 b illustrates theL_(∞) norm difference between the prepared and the target distributionas a function of the digitization mesh size M_(digit) for two differentcircuit depts L and for the n=5 qubit case. Graph 900 c depicted in FIG.9C comprises an example, non-limiting alternative embodiment of graph900 a, where graph 900 c illustrates the L_(∞) norm difference betweenthe prepared and the target distribution as a function of thedigitization mesh size M_(digit) for two different circuit depts L andfor the n=6 qubit case.

FIG. 10 illustrates a flow diagram of an example, non-limitingcomputer-implemented method 1000 that can facilitate estimation ofquantum resources to calculate an expectation value of a stochasticprocess using a re-parameterization method in accordance with one ormore embodiments described herein. Repetitive description of likeelements and/or processes employed in respective embodiments is omittedfor sake of brevity.

At 1002, computer-implemented method 1000 can comprise applying, by asystem (e.g., via quantum resource estimation system 102 and/orre-parameterization component 108) operatively coupled to a processor(e.g., processor 106), a quantum fault-tolerant operation to avariationally prepared quantum state corresponding to a probabilitydistribution to produce a quantum state corresponding to a targetprobability distribution.

At 1004, computer-implemented method 1000 can comprise estimating, bythe system (e.g., via quantum resource estimation system 102 and/orestimation component 110), at least one defined criterion of a quantumcomputer to be used to compute an expectation value of a stochasticprocess (e.g., the value of a derivative asset) associated with thetarget probability distribution.

Quantum resource estimation system 102 can be associated with varioustechnologies. For example, quantum resource estimation system 102 can beassociated with quantum computing technologies, quantum hardware and/orsoftware technologies, quantum algorithm technologies, machine learningtechnologies, artificial intelligence technologies, cloud computingtechnologies, and/or other technologies.

Quantum resource estimation system 102 can provide technicalimprovements to systems, devices, components, operational steps, and/orprocessing steps associated with the various technologies identifiedabove. For example, quantum resource estimation system 102 can: apply aquantum fault-tolerant operation to a variationally prepared quantumstate corresponding to a probability distribution to produce a quantumstate corresponding to a target probability distribution; and/orestimate at least one defined criterion of a quantum computer to be usedto compute an expectation value of a stochastic process (e.g., the valueof a derivative asset) associated with the target probabilitydistribution. In this example, the at least one defined criterion cancomprise an attribute, a condition, a property, a parameter, or aconfiguration of the quantum computer that enables the quantum computerto achieve a defined quantum advantage in computing the expectationvalue of the stochastic process (e.g., the value of a derivative asset)associated with the target probability distribution. In this example,quantum resource estimation system 102 can therefore be implemented toidentify a quantum resource (e.g., a quantum computer, a quantumprocessor, and/or another quantum resource) that can leverage theadvantage of quantum computing to calculate the expectation value of astochastic process (e.g., the value of a derivative asset such as, forinstance, an option contract), while accruing the least amount ofcomputational costs relative to other quantum resources.

Quantum resource estimation system 102 can provide technicalimprovements to a processing unit (e.g., processor 106, a quantumprocessor, and/or another processor) associated with quantum resourceestimation system 102. For example, as described above, quantum resourceestimation system 102 can estimate, and therefore, can also determinethe at least one defined criterion that can enable a quantum computer toleverage the advantage of quantum computing to calculate the expectationvalue of a stochastic process (e.g., the value of a derivative assetsuch as, for instance, an option contract), while accruing the leastamount of computational costs relative to other quantum resources. Inthis example, a quantum processor in a quantum computer can be developed(e.g., engineered, designed, and/or fabricated) and/or modified suchthat it comprises the at least one defined criterion that can beestimated and/or identified by quantum resource estimation system 102 toenable the quantum computer to calculate the expectation value of astochastic process (e.g., the value of a derivative asset such as, forinstance, an option contract), while accruing the least amount ofcomputational costs.

A practical application of quantum resource estimation system 102 isthat it can be implemented using a classical computing device (e.g., aclassical computer) to estimate at least one defined criterion that canenable a quantum computer to leverage the advantage of quantum computingto compute one or more solutions (e.g., heuristic(s)) to a variety ofproblems ranging in complexity (e.g., an estimation problem, anoptimization problem, and/or another problem) in a variety of domains(e.g., finance, chemistry, medicine, and/or another domain). Forexample, a practical application of quantum resource estimation system102 is that it can be implemented using a classical computing device(e.g., a classical computer) to estimate at least one defined criterionthat can enable a quantum computer to leverage the advantage of quantumcomputing to compute one or more solutions (e.g., heuristic(s)) to anestimation problem and/or an optimization problem in the domain ofchemistry, medicine, and/or finance, where such a solution can be usedto engineer, for instance, a new chemical compound, a new medication,and/or a new option premium.

It should be appreciated that quantum resource estimation system 102provides a new approach driven by relatively new quantum computingtechnologies. For example, quantum resource estimation system 102provides a new approach to estimate at least one defined criterion thatcan enable a quantum computer to leverage the advantage of quantumcomputing to calculate an expectation value of a stochastic process(e.g., the value of a derivative asset such as, for instance, an optioncontract), while accruing the least amount of computational costsrelative to other quantum resources.

Quantum resource estimation system 102 can employ hardware or softwareto solve problems that are highly technical in nature, that are notabstract and that cannot be performed as a set of mental acts by ahuman. In some embodiments, one or more of the processes describedherein can be performed by one or more specialized computers (e.g., aspecialized processing unit, a specialized classical computer, aspecialized quantum computer, and/or another type of specializedcomputer) to execute defined tasks related to the various technologiesidentified above. Quantum resource estimation system 102 and/orcomponents thereof, can be employed to solve new problems that arisethrough advancements in technologies mentioned above, employment ofquantum computing systems, cloud computing systems, computerarchitecture, and/or another technology.

It is to be appreciated that quantum resource estimation system 102 canutilize various combinations of electrical components, mechanicalcomponents, and circuitry that cannot be replicated in the mind of ahuman or performed by a human, as the various operations that can beexecuted by quantum resource estimation system 102 and/or componentsthereof as described herein are operations that are greater than thecapability of a human mind. For instance, the amount of data processed,the speed of processing such data, or the types of data processed byquantum resource estimation system 102 over a certain period of time canbe greater, faster, or different than the amount, speed, or data typethat can be processed by a human mind over the same period of time.

According to several embodiments, quantum resource estimation system 102can also be fully operational towards performing one or more otherfunctions (e.g., fully powered on, fully executed, and/or anotherfunction) while also performing the various operations described herein.It should be appreciated that such simultaneous multi-operationalexecution is beyond the capability of a human mind. It should also beappreciated that quantum resource estimation system 102 can includeinformation that is impossible to obtain manually by an entity, such asa human user. For example, the type, amount, and/or variety ofinformation included in quantum resource estimation system 102,re-parameterization component 108, estimation component 110, variationalcomponent 202, and/or error analysis component 302 can be more complexthan information obtained manually by a human user.

In some embodiments, quantum resource estimation system 102 can beassociated with a cloud computing environment. For example, quantumresource estimation system 102 can be associated with cloud computingenvironment 1250 described below with reference to FIG. 12 and/or one ormore functional abstraction layers described below with reference toFIG. 13 (e.g., hardware and software layer 1360, virtualization layer1370, management layer 1380, and/or workloads layer 1390).

Quantum resource estimation system 102 and/or components thereof (e.g.,re-parameterization component 108, estimation component 110, variationalcomponent 202, error analysis component 302, and/or another component)can employ one or more computing resources of cloud computingenvironment 1250 described below with reference to FIG. 12 and/or one ormore functional abstraction layers (e.g., quantum software) describedbelow with reference to FIG. 13 to execute one or more operations inaccordance with one or more embodiments of the subject disclosuredescribed herein. For example, cloud computing environment 1250 and/orsuch one or more functional abstraction layers can comprise one or moreclassical computing devices (e.g., classical computer, classicalprocessor, virtual machine, server, and/or another classical computingdevice), quantum hardware, and/or quantum software (e.g., quantumcomputing device, quantum computer, quantum processor, quantum circuitsimulation software, superconducting circuit, and/or other quantumhardware and/or quantum software) that can be employed by quantumresource estimation system 102 and/or components thereof to execute oneor more operations in accordance with one or more embodiments of thesubject disclosure described herein. For instance, quantum resourceestimation system 102 and/or components thereof can employ such one ormore classical and/or quantum computing resources to execute one or moreclassical and/or quantum: mathematical function, calculation, and/orequation; computing and/or processing script; algorithm; model (e.g.,artificial intelligence (AI) model, machine learning (ML) model, and/oranother type of model); and/or another operation in accordance with oneor more embodiments of the subject disclosure described herein.

It is to be understood that although this disclosure includes a detaileddescription on cloud computing, implementation of the teachings recitedherein are not limited to a cloud computing environment. Rather,embodiments of the present invention are capable of being implemented inconjunction with any other type of computing environment now known orlater developed.

Cloud computing is a model of service delivery for enabling convenient,on-demand network access to a shared pool of configurable computingresources (e.g., networks, network bandwidth, servers, processing,memory, storage, applications, virtual machines, and services) that canbe rapidly provisioned and released with minimal management effort orinteraction with a provider of the service. This cloud model may includeat least five characteristics, at least three service models, and atleast four deployment models.

Characteristics are as follows:

On-demand self-service: a cloud consumer can unilaterally provisioncomputing capabilities, such as server time and network storage, asneeded automatically without requiring human interaction with theservice's provider.

Broad network access: capabilities are available over a network andaccessed through standard mechanisms that promote use by heterogeneousthin or thick client platforms (e.g., mobile phones, laptops, and PDAs).

Resource pooling: the provider's computing resources are pooled to servemultiple consumers using a multi-tenant model, with different physicaland virtual resources dynamically assigned and reassigned according todemand. There is a sense of location independence in that the consumergenerally has no control or knowledge over the exact location of theprovided resources but may be able to specify location at a higher levelof abstraction (e.g., country, state, or datacenter).

Rapid elasticity: capabilities can be rapidly and elasticallyprovisioned, in some cases automatically, to quickly scale out andrapidly released to quickly scale in. To the consumer, the capabilitiesavailable for provisioning often appear to be unlimited and can bepurchased in any quantity at any time.

Measured service: cloud systems automatically control and optimizeresource use by leveraging a metering capability at some level ofabstraction appropriate to the type of service (e.g., storage,processing, bandwidth, and active user accounts). Resource usage can bemonitored, controlled, and reported, providing transparency for both theprovider and consumer of the utilized service.

Service Models are as follows:

Software as a Service (SaaS): the capability provided to the consumer isto use the provider's applications running on a cloud infrastructure.The applications are accessible from various client devices through athin client interface such as a web browser (e.g., web-based e-mail).The consumer does not manage or control the underlying cloudinfrastructure including network, servers, operating systems, storage,or even individual application capabilities, with the possible exceptionof limited user-specific application configuration settings.

Platform as a Service (PaaS): the capability provided to the consumer isto deploy onto the cloud infrastructure consumer-created or acquiredapplications created using programming languages and tools supported bythe provider. The consumer does not manage or control the underlyingcloud infrastructure including networks, servers, operating systems, orstorage, but has control over the deployed applications and possiblyapplication hosting environment configurations.

Infrastructure as a Service (IaaS): the capability provided to theconsumer is to provision processing, storage, networks, and otherfundamental computing resources where the consumer is able to deploy andrun arbitrary software, which can include operating systems andapplications. The consumer does not manage or control the underlyingcloud infrastructure but has control over operating systems, storage,deployed applications, and possibly limited control of select networkingcomponents (e.g., host firewalls).

Deployment Models are as follows:

Private cloud: the cloud infrastructure is operated solely for anorganization. It may be managed by the organization or a third party andmay exist on-premises or off-premises.

Community cloud: the cloud infrastructure is shared by severalorganizations and supports a specific community that has shared concerns(e.g., mission, security requirements, policy, and complianceconsiderations). It may be managed by the organizations or a third partyand may exist on-premises or off-premises.

Public cloud: the cloud infrastructure is made available to the generalpublic or a large industry group and is owned by an organization sellingcloud services.

Hybrid cloud: the cloud infrastructure is a composition of two or moreclouds (private, community, or public) that remain unique entities butare bound together by standardized or proprietary technology thatenables data and application portability (e.g., cloud bursting forload-balancing between clouds).

A cloud computing environment is service oriented with a focus onstatelessness, low coupling, modularity, and semantic interoperability.At the heart of cloud computing is an infrastructure that includes anetwork of interconnected nodes.

For simplicity of explanation, the computer-implemented methodologiesare depicted and described as a series of acts. It is to be understoodand appreciated that the subject innovation is not limited by the actsillustrated and/or by the order of acts, for example acts can occur invarious orders and/or concurrently, and with other acts not presentedand described herein. Furthermore, not all illustrated acts can berequired to implement the computer-implemented methodologies inaccordance with the disclosed subject matter. In addition, those skilledin the art will understand and appreciate that the computer-implementedmethodologies can alternatively be represented as a series ofinterrelated states via a state diagram or events. Additionally, itshould be further appreciated that the computer-implementedmethodologies disclosed hereinafter and throughout this specificationare capable of being stored on an article of manufacture to facilitatetransporting and transferring such computer-implemented methodologies tocomputers. The term article of manufacture, as used herein, is intendedto encompass a computer program accessible from any computer-readabledevice or storage media.

In order to provide a context for the various aspects of the disclosedsubject matter, FIG. 11 as well as the following discussion are intendedto provide a general description of a suitable environment in which thevarious aspects of the disclosed subject matter can be implemented. FIG.11 illustrates a block diagram of an example, non-limiting operatingenvironment in which one or more embodiments described herein can befacilitated. Repetitive description of like elements employed in otherembodiments described herein is omitted for sake of brevity.

With reference to FIG. 11, a suitable operating environment 1100 forimplementing various aspects of this disclosure can also include acomputer 1112. The computer 1112 can also include a processing unit1114, a system memory 1116, and a system bus 1118. The system bus 1118couples system components including, but not limited to, the systemmemory 1116 to the processing unit 1114. The processing unit 1114 can beany of various available processors. Dual microprocessors and othermultiprocessor architectures also can be employed as the processing unit1114. The system bus 1118 can be any of several types of busstructure(s) including the memory bus or memory controller, a peripheralbus or external bus, and/or a local bus using any variety of availablebus architectures including, but not limited to, Industrial StandardArchitecture (ISA), Micro-Channel Architecture (MSA), Extended ISA(EISA), Intelligent Drive Electronics (IDE), VESA Local Bus (VLB),Peripheral Component Interconnect (PCI), Card Bus, Universal Serial Bus(USB), Advanced Graphics Port (AGP), Firewire (IEEE 1394), and SmallComputer Systems Interface (SCSI).

The system memory 1116 can also include volatile memory 1120 andnonvolatile memory 1122. The basic input/output system (BIOS),containing the basic routines to transfer information between elementswithin the computer 1112, such as during start-up, is stored innonvolatile memory 1122. Computer 1112 can also includeremovable/non-removable, volatile/non-volatile computer storage media.FIG. 11 illustrates, for example, a disk storage 1124. Disk storage 1124can also include, but is not limited to, devices like a magnetic diskdrive, floppy disk drive, tape drive, Jaz drive, Zip drive, LS-100drive, flash memory card, or memory stick. The disk storage 1124 alsocan include storage media separately or in combination with otherstorage media. To facilitate connection of the disk storage 1124 to thesystem bus 1118, a removable or non-removable interface is typicallyused, such as interface 1126. FIG. 11 also depicts software that acts asan intermediary between users and the basic computer resources describedin the suitable operating environment 1100. Such software can alsoinclude, for example, an operating system 1128. Operating system 1128,which can be stored on disk storage 1124, acts to control and allocateresources of the computer 1112.

System applications 1130 take advantage of the management of resourcesby operating system 1128 through program modules 1132 and program data1134, e.g., stored either in system memory 1116 or on disk storage 1124.It is to be appreciated that this disclosure can be implemented withvarious operating systems or combinations of operating systems. A userenters commands or information into the computer 1112 through inputdevice(s) 1136. Input devices 1136 include, but are not limited to, apointing device such as a mouse, trackball, stylus, touch pad, keyboard,microphone, joystick, game pad, satellite dish, scanner, TV tuner card,digital camera, digital video camera, web camera, and the like. Theseand other input devices connect to the processing unit 1114 through thesystem bus 1118 via interface port(s) 1138. Interface port(s) 1138include, for example, a serial port, a parallel port, a game port, and auniversal serial bus (USB). Output device(s) 1140 use some of the sametype of ports as input device(s) 1136. Thus, for example, a USB port canbe used to provide input to computer 1112, and to output informationfrom computer 1112 to an output device 1140. Output adapter 1142 isprovided to illustrate that there are some output devices 1140 likemonitors, speakers, and printers, among other output devices 1140, whichcan require special adapters. The output adapters 1142 include, by wayof illustration and not limitation, video and sound cards that provide ameans of connection between the output device 1140 and the system bus1118. It should be noted that other devices and/or systems of devicesprovide both input and output capabilities such as remote computer(s)1144.

Computer 1112 can operate in a networked environment using logicalconnections to one or more remote computers, such as remote computer(s)1144. The remote computer(s) 1144 can be a computer, a server, a router,a network PC, a workstation, a microprocessor based appliance, a peerdevice or other common network node and the like, and typically can alsoinclude many or all of the elements described relative to computer 1112.For purposes of brevity, only a memory storage device 1146 isillustrated with remote computer(s) 1144. Remote computer(s) 1144 islogically connected to computer 1112 through a network interface 1148and then physically connected via communication connection 1150. Networkinterface 1148 encompasses wire and/or wireless communication networkssuch as local-area networks (LAN), wide-area networks (WAN), cellularnetworks, and/or another wire and/or wireless communication network. LANtechnologies include Fiber Distributed Data Interface (FDDI), CopperDistributed Data Interface (CDDI), Ethernet, Token Ring and the like.WAN technologies include, but are not limited to, point-to-point links,circuit switching networks like Integrated Services Digital Networks(ISDN) and variations thereon, packet switching networks, and DigitalSubscriber Lines (DSL). Communication connection(s) 1150 refers to thehardware/software employed to connect the network interface 1148 to thesystem bus 1118. While communication connection 1150 is shown forillustrative clarity inside computer 1112, it can also be external tocomputer 1112. The hardware/software for connection to the networkinterface 1148 can also include, for exemplary purposes only, internaland external technologies such as, modems including regular telephonegrade modems, cable modems and DSL modems, ISDN adapters, and Ethernetcards.

Referring now to FIG. 12, an illustrative cloud computing environment1250 is depicted. As shown, cloud computing environment 1250 includesone or more cloud computing nodes 1210 with which local computingdevices used by cloud consumers, such as, for example, personal digitalassistant (PDA) or cellular telephone 1254A, desktop computer 1254B,laptop computer 1254C, and/or automobile computer system 1254N maycommunicate. Although not illustrated in FIG. 12, cloud computing nodes1210 can further comprise a quantum platform (e.g., quantum computer,quantum hardware, quantum software, and/or another quantum platform)with which local computing devices used by cloud consumers cancommunicate. Nodes 1210 may communicate with one another. They may begrouped (not shown) physically or virtually, in one or more networks,such as Private, Community, Public, or Hybrid clouds as describedhereinabove, or a combination thereof. This allows cloud computingenvironment 1250 to offer infrastructure, platforms and/or software asservices for which a cloud consumer does not need to maintain resourceson a local computing device. It is understood that the types ofcomputing devices 1254A-N shown in FIG. 12 are intended to beillustrative only and that computing nodes 1210 and cloud computingenvironment 1250 can communicate with any type of computerized deviceover any type of network and/or network addressable connection (e.g.,using a web browser).

Referring now to FIG. 13, a set of functional abstraction layersprovided by cloud computing environment 1250 (FIG. 12) is shown. Itshould be understood in advance that the components, layers, andfunctions shown in FIG. 13 are intended to be illustrative only andembodiments of the invention are not limited thereto. As depicted, thefollowing layers and corresponding functions are provided:

Hardware and software layer 1360 includes hardware and softwarecomponents. Examples of hardware components include: mainframes 1361;RISC (Reduced Instruction Set Computer) architecture based servers 1362;servers 1363; blade servers 1364; storage devices 1365; and networks andnetworking components 1366. In some embodiments, software componentsinclude network application server software 1367, database software1368, quantum platform routing software (not illustrated in FIG. 13),and/or quantum software (not illustrated in FIG. 13).

Virtualization layer 1370 provides an abstraction layer from which thefollowing examples of virtual entities may be provided: virtual servers1371; virtual storage 1372; virtual networks 1373, including virtualprivate networks; virtual applications and operating systems 1374; andvirtual clients 1375.

In one example, management layer 1380 may provide the functionsdescribed below. Resource provisioning 1381 provides dynamic procurementof computing resources and other resources that are utilized to performtasks within the cloud computing environment. Metering and Pricing 1382provide cost tracking as resources are utilized within the cloudcomputing environment, and billing or invoicing for consumption of theseresources. In one example, these resources may include applicationsoftware licenses. Security provides identity verification for cloudconsumers and tasks, as well as protection for data and other resources.User portal 1383 provides access to the cloud computing environment forconsumers and system administrators. Service level management 1384provides cloud computing resource allocation and management such thatrequired service levels are met. Service Level Agreement (SLA) planningand fulfillment 1385 provide pre-arrangement for, and procurement of,cloud computing resources for which a future requirement is anticipatedin accordance with an SLA.

Workloads layer 1390 provides examples of functionality for which thecloud computing environment may be utilized. Non-limiting examples ofworkloads and functions which may be provided from this layer include:mapping and navigation 1391; software development and lifecyclemanagement 1392; virtual classroom education delivery 1393; dataanalytics processing 1394; transaction processing 1395; and quantumresource estimation software 1396.

The present invention may be a system, a method, an apparatus and/or acomputer program product at any possible technical detail level ofintegration. The computer program product can include a computerreadable storage medium (or media) having computer readable programinstructions thereon for causing a processor to carry out aspects of thepresent invention. The computer readable storage medium can be atangible device that can retain and store instructions for use by aninstruction execution device. The computer readable storage medium canbe, for example, but is not limited to, an electronic storage device, amagnetic storage device, an optical storage device, an electromagneticstorage device, a semiconductor storage device, or any suitablecombination of the foregoing. A non-exhaustive list of more specificexamples of the computer readable storage medium can also include thefollowing: a portable computer diskette, a hard disk, a random accessmemory (RAM), a read-only memory (ROM), an erasable programmableread-only memory (EPROM or Flash memory), a static random access memory(SRAM), a portable compact disc read-only memory (CD-ROM), a digitalversatile disk (DVD), a memory stick, a floppy disk, a mechanicallyencoded device such as punch-cards or raised structures in a groovehaving instructions recorded thereon, and any suitable combination ofthe foregoing. A computer readable storage medium, as used herein, isnot to be construed as being transitory signals per se, such as radiowaves or other freely propagating electromagnetic waves, electromagneticwaves propagating through a waveguide or other transmission media (e.g.,light pulses passing through a fiber-optic cable), or electrical signalstransmitted through a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network can comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device. Computer readable programinstructions for carrying out operations of the present invention can beassembler instructions, instruction-set-architecture (ISA) instructions,machine instructions, machine dependent instructions, microcode,firmware instructions, state-setting data, configuration data forintegrated circuitry, or either source code or object code written inany combination of one or more programming languages, including anobject oriented programming language such as Smalltalk, C++, or thelike, and procedural programming languages, such as the “C” programminglanguage or similar programming languages. The computer readable programinstructions can execute entirely on the user's computer, partly on theuser's computer, as a stand-alone software package, partly on the user'scomputer and partly on a remote computer or entirely on the remotecomputer or server. In the latter scenario, the remote computer can beconnected to the user's computer through any type of network, includinga local area network (LAN) or a wide area network (WAN), or theconnection can be made to an external computer (for example, through theInternet using an Internet Service Provider). In some embodiments,electronic circuitry including, for example, programmable logiccircuitry, field-programmable gate arrays (FPGA), or programmable logicarrays (PLA) can execute the computer readable program instructions byutilizing state information of the computer readable programinstructions to personalize the electronic circuitry, in order toperform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions. These computer readable programinstructions can be provided to a processor of a general purposecomputer, special purpose computer, or other programmable dataprocessing apparatus to produce a machine, such that the instructions,which execute via the processor of the computer or other programmabledata processing apparatus, create means for implementing thefunctions/acts specified in the flowchart and/or block diagram block orblocks. These computer readable program instructions can also be storedin a computer readable storage medium that can direct a computer, aprogrammable data processing apparatus, and/or other devices to functionin a particular manner, such that the computer readable storage mediumhaving instructions stored therein comprises an article of manufactureincluding instructions which implement aspects of the function/actspecified in the flowchart and/or block diagram block or blocks. Thecomputer readable program instructions can also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational acts to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams can represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks can occur out of theorder noted in the Figures. For example, two blocks shown in successioncan, in fact, be executed substantially concurrently, or the blocks cansometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

While the subject matter has been described above in the general contextof computer-executable instructions of a computer program product thatruns on a computer and/or computers, those skilled in the art willrecognize that this disclosure also can or can be implemented incombination with other program modules. Generally, program modulesinclude routines, programs, components, data structures, and/or otherprogram modules that perform particular tasks and/or implementparticular abstract data types. Moreover, those skilled in the art willappreciate that the inventive computer-implemented methods can bepracticed with other computer system configurations, includingsingle-processor or multiprocessor computer systems, mini-computingdevices, mainframe computers, as well as computers, hand-held computingdevices (e.g., PDA, phone), microprocessor-based or programmableconsumer or industrial electronics, and the like. The illustratedaspects can also be practiced in distributed computing environments inwhich tasks are performed by remote processing devices that are linkedthrough a communications network. However, some, if not all aspects ofthis disclosure can be practiced on stand-alone computers. In adistributed computing environment, program modules can be located inboth local and remote memory storage devices. For example, in one ormore embodiments, computer executable components can be executed frommemory that can include or be comprised of one or more distributedmemory units. As used herein, the term “memory” and “memory unit” areinterchangeable. Further, one or more embodiments described herein canexecute code of the computer executable components in a distributedmanner, e.g., multiple processors combining or working cooperatively toexecute code from one or more distributed memory units. As used herein,the term “memory” can encompass a single memory or memory unit at onelocation or multiple memories or memory units at one or more locations.

As used in this application, the terms “component,” “system,”“platform,” “interface,” and the like, can refer to and/or can include acomputer-related entity or an entity related to an operational machinewith one or more specific functionalities. The entities disclosed hereincan be either hardware, a combination of hardware and software,software, or software in execution. For example, a component can be, butis not limited to being, a process running on a processor, a processor,an object, an executable, a thread of execution, a program, and/or acomputer. By way of illustration, both an application running on aserver and the server can be a component. One or more components canreside within a process and/or thread of execution and a component canbe localized on one computer and/or distributed between two or morecomputers. In another example, respective components can execute fromvarious computer readable media having various data structures storedthereon. The components can communicate via local and/or remoteprocesses such as in accordance with a signal having one or more datapackets (e.g., data from one component interacting with anothercomponent in a local system, distributed system, and/or across a networksuch as the Internet with other systems via the signal). As anotherexample, a component can be an apparatus with specific functionalityprovided by mechanical parts operated by electric or electroniccircuitry, which is operated by a software or firmware applicationexecuted by a processor. In such a case, the processor can be internalor external to the apparatus and can execute at least a part of thesoftware or firmware application. As yet another example, a componentcan be an apparatus that provides specific functionality throughelectronic components without mechanical parts, wherein the electroniccomponents can include a processor or other means to execute software orfirmware that confers at least in part the functionality of theelectronic components. In an aspect, a component can emulate anelectronic component via a virtual machine, e.g., within a cloudcomputing system.

In addition, the term “or” is intended to mean an inclusive “or” ratherthan an exclusive “or.” That is, unless specified otherwise, or clearfrom context, “X employs A or B” is intended to mean any of the naturalinclusive permutations. That is, if X employs A; X employs B; or Xemploys both A and B, then “X employs A or B” is satisfied under any ofthe foregoing instances. Moreover, articles “a” and “an” as used in thesubject specification and annexed drawings should generally be construedto mean “one or more” unless specified otherwise or clear from contextto be directed to a singular form. As used herein, the terms “example”and/or “exemplary” are utilized to mean serving as an example, instance,or illustration. For the avoidance of doubt, the subject matterdisclosed herein is not limited by such examples. In addition, anyaspect or design described herein as an “example” and/or “exemplary” isnot necessarily to be construed as preferred or advantageous over otheraspects or designs, nor is it meant to preclude equivalent exemplarystructures and techniques known to those of ordinary skill in the art.

As it is employed in the subject specification, the term “processor” canrefer to substantially any computing processing unit or devicecomprising, but not limited to, single-core processors;single-processors with software multithread execution capability;multi-core processors; multi-core processors with software multithreadexecution capability; multi-core processors with hardware multithreadtechnology; parallel platforms; and parallel platforms with distributedshared memory. Additionally, a processor can refer to an integratedcircuit, an application specific integrated circuit (ASIC), a digitalsignal processor (DSP), a field programmable gate array (FPGA), aprogrammable logic controller (PLC), a complex programmable logic device(CPLD), a discrete gate or transistor logic, discrete hardwarecomponents, or any combination thereof designed to perform the functionsdescribed herein. Further, processors can exploit nano-scalearchitectures such as, but not limited to, molecular and quantum-dotbased transistors, switches and gates, in order to optimize space usageor enhance performance of user equipment. A processor can also beimplemented as a combination of computing processing units. In thisdisclosure, terms such as “store,” “storage,” “data store,” datastorage,” “database,” and substantially any other information storagecomponent relevant to operation and functionality of a component areutilized to refer to “memory components,” entities embodied in a“memory,” or components comprising a memory. It is to be appreciatedthat memory and/or memory components described herein can be eithervolatile memory or nonvolatile memory, or can include both volatile andnonvolatile memory. By way of illustration, and not limitation,nonvolatile memory can include read only memory (ROM), programmable ROM(PROM), electrically programmable ROM (EPROM), electrically erasable ROM(EEPROM), flash memory, or nonvolatile random access memory (RAM) (e.g.,ferroelectric RAM (FeRAM). Volatile memory can include RAM, which canact as external cache memory, for example. By way of illustration andnot limitation, RAM is available in many forms such as synchronous RAM(SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rateSDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM),direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), andRambus dynamic RAM (RDRAM). Additionally, the disclosed memorycomponents of systems or computer-implemented methods herein areintended to include, without being limited to including, these and anyother suitable types of memory.

What has been described above include mere examples of systems andcomputer-implemented methods. It is, of course, not possible to describeevery conceivable combination of components or computer-implementedmethods for purposes of describing this disclosure, but one of ordinaryskill in the art can recognize that many further combinations andpermutations of this disclosure are possible. Furthermore, to the extentthat the terms “includes,” “has,” “possesses,” and the like are used inthe detailed description, claims, appendices and drawings such terms areintended to be inclusive in a manner similar to the term “comprising” as“comprising” is interpreted when employed as a transitional word in aclaim.

The descriptions of the various embodiments have been presented forpurposes of illustration, but are not intended to be exhaustive orlimited to the embodiments disclosed. Many modifications and variationswill be apparent to those of ordinary skill in the art without departingfrom the scope and spirit of the described embodiments. The terminologyused herein was chosen to best explain the principles of theembodiments, the practical application or technical improvement overtechnologies found in the marketplace, or to enable others of ordinaryskill in the art to understand the embodiments disclosed herein.

What is claimed is:
 1. A system, comprising: a processor that executescomputer executable components stored in memory, the computer executablecomponents comprising: a re-parameterization component that applies aquantum fault-tolerant operation to a variationally prepared quantumstate corresponding to a probability distribution to produce a quantumstate corresponding to a target probability distribution; and anestimation component that estimates at least one defined criterion of aquantum computer to be used to compute an expectation value of astochastic process associated with the target probability distribution.2. The system of claim 1, wherein the re-parameterization componentapplies a transformation operation to the variationally prepared quantumstate to produce the quantum state corresponding to the targetprobability distribution with at least one of: a defined mean of thetarget probability distribution; a defined standard deviation of thetarget probability distribution; or one or more explicit parameters thatspecify the target probability distribution.
 3. The system of claim 1,wherein the re-parameterization component applies the quantumfault-tolerant operation to the variationally prepared quantum state toprepare the quantum state as a superposition over possible paths of adiscrete time multivariate stochastic process.
 4. The system of claim 1,wherein the computer executable components further comprise: avariational component that trains a variational quantum circuit toprepare the variationally prepared quantum state and to reducecomputational costs of quantum arithmetic operations performed by thequantum computer to compute the expectation value of the stochasticprocess associated with the target probability distribution.
 5. Thesystem of claim 1, wherein the computer executable components furthercomprise: a variational component that trains a variational quantumcircuit to prepare the variationally prepared quantum state, and whereinthe variational component trains the variational quantum circuit using aHamiltonian operator to generate a ground state corresponding to thetarget probability distribution.
 6. The system of claim 1, wherein thecomputer executable components further comprise: an error analysiscomponent that calculates one or more errors associated with at leastone of: application of the quantum fault-tolerant operation to thevariationally prepared quantum state to produce the quantum state;estimation of the at least one defined criterion; or computation of theexpectation value of the stochastic process associated with the targetprobability distribution.
 7. The system of claim 1, wherein the at leastone defined criterion is selected from a group consisting of anattribute, a condition, a property, a parameter, or a configuration ofthe quantum computer that enables the quantum computer to achieve adefined quantum advantage in computing the expectation value of thestochastic process associated with the target probability distribution,and wherein the probability distribution comprises a standard normalprobability distribution and the target probability distributioncomprises a normal probability distribution.
 8. A computer-implementedmethod, comprising: applying, by a system operatively coupled to aprocessor, a quantum fault-tolerant operation to a variationallyprepared quantum state corresponding to a probability distribution toproduce a quantum state corresponding to a target probabilitydistribution; and estimating, by the system, at least one definedcriterion of a quantum computer to be used to compute an expectationvalue of a stochastic process associated with the target probabilitydistribution.
 9. The computer-implemented method of claim 8, furthercomprising: applying, by the system, a transformation operation to thevariationally prepared quantum state to produce the quantum statecorresponding to the target probability distribution with at least oneof: a defined mean of the target probability distribution; a definedstandard deviation of the target probability distribution; or one ormore explicit parameters that specify the target probabilitydistribution.
 10. The computer-implemented method of claim 8, furthercomprising: applying, by the system, the quantum fault-tolerantoperation to the variationally prepared quantum state to prepare thequantum state as a superposition over possible paths of a discrete timemultivariate stochastic process.
 11. The computer-implemented method ofclaim 8, further comprising: training, by the system, a variationalquantum circuit to prepare the variationally prepared quantum state andto reduce computational costs of quantum arithmetic operations performedby the quantum computer to compute the expectation value of thestochastic process associated with the target probability distribution.12. The computer-implemented method of claim 8, further comprising:training, by the system, a variational quantum circuit to prepare thevariationally prepared quantum state; and training, by the system, thevariational quantum circuit using a Hamiltonian operator to generate aground state corresponding to the target probability distribution. 13.The computer-implemented method of claim 8, further comprising:calculating, by the system, one or more errors associated with at leastone of: application of the quantum fault-tolerant operation to thevariationally prepared quantum state to produce the quantum state;estimation of the at least one defined criterion; or computation of theexpectation value of the stochastic process associated with the targetprobability distribution.
 14. The computer-implemented method of claim8, wherein the at least one defined criterion is selected from a groupconsisting of an attribute, a condition, a property, a parameter, or aconfiguration of the quantum computer that enables the quantum computerto achieve a defined quantum advantage in computing the expectationvalue of the stochastic process associated with the target probabilitydistribution, and wherein the probability distribution comprises astandard normal probability distribution and the target probabilitydistribution comprises a normal probability distribution.
 15. A computerprogram product comprising a computer readable storage medium havingprogram instructions embodied therewith, the program instructionsexecutable by a processor to cause the processor to: apply a quantumfault-tolerant operation to a variationally prepared quantum statecorresponding to a probability distribution to produce a quantum statecorresponding to a target probability distribution; and estimate atleast one defined criterion of a quantum computer to be used to computean expectation value of a stochastic process associated with the targetprobability distribution.
 16. The computer program product of claim 15,wherein the program instructions are further executable by the processorto cause the processor to: apply a transformation operation to thevariationally prepared quantum state to produce the quantum statecorresponding to the target probability distribution with at least oneof: a defined mean of the target probability distribution; a definedstandard deviation of the target probability distribution; or one ormore explicit parameters that specify the target probabilitydistribution.
 17. The computer program product of claim 15, wherein theprogram instructions are further executable by the processor to causethe processor to: apply the quantum fault-tolerant operation to thevariationally prepared quantum state to prepare the quantum state as asuperposition over possible paths of a discrete time multivariatestochastic process.
 18. The computer program product of claim 15,wherein the program instructions are further executable by the processorto cause the processor to: train a variational quantum circuit toprepare the variationally prepared quantum state and to reducecomputational costs of quantum arithmetic operations performed by thequantum computer to compute the expectation value of the stochasticprocess associated with the target probability distribution.
 19. Thecomputer program product of claim 15, wherein the program instructionsare further executable by the processor to cause the processor to: traina variational quantum circuit to prepare the variationally preparedquantum state; and train the variational quantum circuit using aHamiltonian operator to generate a ground state corresponding to thetarget probability distribution.
 20. The computer program product ofclaim 15, wherein the program instructions are further executable by theprocessor to cause the processor to: calculate one or more errorsassociated with at least one of: application of the quantumfault-tolerant operation to the variationally prepared quantum state toproduce the quantum state; estimation of the at least one definedcriterion; or computation of the expectation value of the stochasticprocess associated with the target probability distribution.